I am not familiar with the D750 sound, and can't say whether yours is typical either, but I think you'll find that any SLR will be pretty noisy, because of the mirror movement, and some even without that can have a pretty noisy shutter. I tested my D7100 with a DB meter and found that close up in LV, the sound level drops from about 85 Dba to 80, but makes a longer noise, since the shutter must close, then open and close, and the open again.
Back in the days of the old Nikon F, one could lock up the mirror and get a quiet shutter, but nowadays in "MUP" mode, the mirror drops at the end of each shot, so you get the noise again.
But if you have access to a photo store or other photographers whose gear you can hear, you might want to sample the sound from a DX camera like the D500 or the D7200 (or 7500) , and see if it's a little quieter, because it has a smaller mirror and shutter. It will still make a noise, but it might be a little better. For wildlife a DX is pretty nice, and I hear the latest ones focus nice and fast.
If budget is a constraint, I suspect your best bet is to sneak up on your prey as if you were shooting slides and try for one good, well composed shot, assuming you'll only get one. Or set it on Continuous High and hope for a lucky takeoff shot.
The thought of the shutterless Z9 has me drooling, but aside from its enormous cost and limited availability, it's too big for my habit of international travel with carry-on baggage only, for which even the D7100 is a bit bulky, so no go until they get a little one. A high-zoot shutterless DX, though, and I might just pawn the car.