I think the major problem is that religions are ill-formed. They try to be all-encompassing, so they try to "muscle in" on science, morality, etc. But simply because most religions are ill-formed, and stretch beyond the theological doesn't mean religion in general can't coexist with science, morality, etc. and not interfere with them.
That's a fair point, but then what kind of religion could exist that didn't muscle in on science?

Presumeably such a religion would not make any claims over any phenomena after the big bang.

So God answering prayers and performing miracles? Human beings having a soul?

You would have to lose all those because they are tangible claims over things that are actually supposed to happen in the physical universe, that's science's field.