The question of the compatibility of religion and science depends on what you generally mean by compatible, and religion. But, we'll tend to gloss over compatibilism, but depending on what is meant by those terms it's either true or false, and depends.
You assert that both science and God are essential to our understanding. That already assumes God is real. But further declares God is essential to our understanding. In principle and practice, you actually have to prove that step. And if you did, it would already be a reasonable proof.
The worst problem I see is pretty central to the argument. You seem to be misconstruing being wrong with non-overlapping magisterium. You very much can use science to prove religious claims. But, let me focus down on why you say you can't.
Often times when God is given as an explanation for a natural process it is a gods of the gaps argument. It sees an opening in our understanding (we haven't explained morality, or design in nature, or the origin of the universe) and shoves God in there. Then science comes along and actually gives the correct answer and explains the phenomenon. Using this background you then veer completely the wrong direction. Saying, more or less, that if such points are that way falsified, is it not therefore true that science cannot falsify such notions.
In reality, the way this works is the answer that fails the most, is demonstratively not a good answer. If you had two horses that ran a million races and every time they ran a race the first horse won, and they were preparing to run another race, which horse should we bet on? This is the way it has been with the gods of the gaps, and the track record of the success of supernatural and divine claims. Lighting is caused by Zeus. Wait, it's just a natural phenomenon. Thunder is caused by Thor's Hammer. Wait it's an exploding sonic boom of air around lightning. Disease is caused by demons, and Jesus can shoo those demons away. Wait, they are cause by microscopic life forms. Genetics must be caused by God. Nope. Life must be God by God. Nope. The universe must be caused by God. Nope. -- Each time, supernaturalism is offered as an answer, and each time the answer turns out to be unknown natural phenomenon.
In every case, thus far the answer to every mystery has always turned out to be unknown natural phenomenon. You state in your conclusion that, "to rely solely on science is irrational" -- I must ask you to substantiate that premise. Because, thus far, with regard to how the universe works, the only thing that has ever answered a question, that we use to not know the answer to and we now do know the answer to, made that transition was science. -- At best religion was only interjected into those questions as a placeholder for our ignorance. We don't know why the seasons happen or why there are stars, the gods must have given us the seasons and the stars must be lights embedded in the dome that covers the Earth. -- Now we do. Religion was the wrong answer. And to the best I can tell has never actually answered any question. Science on the other hand has answered everything we've actually answered.
As for the why and how dichotomy, this generally isn't your argument so I'll be a little less kid gloves. It's entirely fictitious. Why are we here? There's plenty of answers. Generally our parents had sex is a pretty good answer to that. How does that work? Well, we have a pretty complete understanding of reproduction. Why does life exist here on this planet ultimately? Because it can. We can't accept all answers. Or rather, we can but we need to check how reasonable they are and decide which one is the most likely to be correct (or which is the least wrong).
The question here isn't really the "most convincing" but actually which one is the most likely to be true. Humans are an odd lot. We are easily distracted by shiny things and hollow promises. So, anybody might be find themselves liking the answer that they are the very special creation of the creator of the universe, who due to accepting these particular beliefs, will be given an eternity of bliss much more than they like the answer, that they evolved on this planet, their parents had sex, and they will die and be dead. While there's nothing to support the former, there's every reason to think the latter is true. But, again, humans love shiny things and hollow promises.
Looking more specifically towards the Christian answers they seem even less convincing. Not because the promises of eternal life and avoiding eternal punishment makes it a sucker's bet to refuse. But, because that stuff is exactly what humans would come up. God needed blood to fix the universe, and only his own blood had enough magic to do it, so he gave himself a body and killed it. He did this to create a loophole in the system he had already invented that nobody could succeed because some mythological ancestors ate magical fruit from a magical tree, and God's anger at them carried down to their innocent children. And finally the only way that God could stop being angry at his creations and needing to send them eternally to the place of infinite torture was to don a baby suit, live a perfect life, and die horribly. Because blood appeases God, and the better the blood, the more it appeases God. So God's blood appeases God the most, and so that was the only way God could be so very appeased to allow all you sinners into heaven anyway, even though you are all vile evil crap. God loved you enough to have a very bad weekend for your sins.
I know I'm making light of the religion there, but if you look at it sociologically you can see all the very human hooks. Humans want to avoid pain and achieve pleasure. So they are promised eternal bliss and to avoid eternal torture. Humans are easily shamed into doing things and guilted into them, so look what you made me do, Jesus had to die for your sins! Or historically it it makes perfect sense that in first century Israel. You have a society that actually believed that blood sacrifice appeased God, and they were very recently taken over by Pagan Rome. And the Pagans absolutely loved their stories about the sons and daughters of gods. Demigods were a hugely popular genre in their culture. And where these two intersect we find the first instances of the religion that says that the Son of God was sacrificed to God for the sins of the people and believing and accepting this gives you eternal life. It later gets complicated through trinitarianism (if the Son of God is God, he sacrificed Himself to Himself), but it's all a pretty direct path to how humans actually invented such things and how the religion was much more strongly influenced by culture than by reality.
If that were true, it would be rather heterodox in and of itself. That seems like a really strange way to run a universe with 70 sextillion stars and hundred of sextillions of planets. But, that humans made up such gods, seems really really likely. Because humans all over this planet make up all kinds of gods. Gods with several heads, that look like humans, that create the universe, that fight and kill giants, that create themselves. We humans love to make up gods. We are easily convinced by such stories, but not because the stories are true. But because the stories are so very deeply human. They are exactly the sorts of things that should convince us, they are specifically designed (self-designed infact) to get followers and be effective, and get more followers and be even more effective.
This is the principle distinction between "convincing" and "correct". We humans can easily be bamboozled, and nobody does it better than we do it to ourselves. Science is so very effective because it pulls out all the stops in order to prevent this from happening. In science, we try very very hard to keep from fooling ourselves.
Science answers the how questions, but it also answers the why questions. Religion answers neither type of question. In the history of answering questions, it's boasting a whopping zero percent. There are not always answers to the why questions. "Why the seas are boiling hot?" "Why were the last powerball numbers 28 36 40 48 55 and 1, and not something else?" However, there's also some how questions that equally don't have answers. "How is the decay of a radioactive isotope caused?" -- Questions do not have to have answers. But, for the questions that don't have answers there's a lot of them that are why questions because it asks for motivations and a lot of things aren't motivated. "Why does the sun rise in the morning?" Sure, inertia and rotating masses, but "Why is it like that?" You can eventually get down to what is basically just statistical noise. Where there is no good reason why it is that and not something else.
And religion comes along, and does what it always does. It shoves god into any question that isn't answered.
However, the core of your argument is that both science and God are essential to our understanding reality. But, is that actually true? What if I insisted that fairies are required to answer these why questions. Any question about why something is that way and not another is attributed to the will of the fairies who made it that way and certainly wouldn't have made it a different way. -- If this even makes logical sense, it cannot be that God is actually essential. God is again, as always, just being inserted as a placeholder for not having an answer, a role indistinguishably similar to my hypothetical fairies. And what if we went the other way, and tried replacing science. That both fishing and God are essential to understanding the universe. We instantly notice that fishing sticks out like a sore thumb. It doesn't ever answer questions, it's a procedure by which we use fishing poles to catch fish.
It seems that we could very well simply reduce that statement to just science. Science is essential to our understanding of reality. And if you want to add to that, you'd need to show that whatever you're adding is capable of solving actual problems and has a track record for having solved actual problems without science. To the best of my knowledge, only mathematics possibly fits that criteria. Why is it that "to rely solely on science is irrational" -- it seems to be the only thing that gives us correct answers, builds computers, takes us to the moon, cures diseases, doubles our lifespan, and actually replace ignorance with understanding.