Hide glue for very dry climates, fish bladder for more humid areas. It does make a difference 100%.
EA40 is stronger when mixed 2:1 and cured at 40 degrees. Its heat resistance definitely goes up too. It will of course hold horn to a core...but I see it as a bit of a cop out when making a hornbow - I like them to be traditional - to each his own. Importantly the bond between horn and core is one of the last to be affected by rain/humidity. the sinew is the weak link with reference to humidity. Often the belly of hornbows wasn't even covered or finished - no need. The painting and leather covering are simply to protect the back (and maybe the edges a little).
I still think the issues with these bows will be stack.
You mention adding a little more reflex. If you don't pull the tips together when adding layers after gelling, before any drying takes place you risk the sinew delaming. Personally if I made this sort of bow I would want the tips touching. The horn and sinew can take it easily. String tension at brace is one of the main reasons hornbows shine. You can't have too much reflex!
Good luck.