I'm not disputing that RX100 series cameras take great images, but the shutter lag, particularly when trying to shoot a moving subject, can be infuriating. I dipped my toes into UW photography with a rented Panasonic DMC-FX35, and compared to that, my Sony A6300 is simply a joy to shoot - and it didn't cost me that much more than your quoted number of $750; I paid $837 for the camera with kit lens and $251 for the Meikon/SeaFrogs housing.
I am prepared to believe that the Panasonic you rented has issues with shutter lag. All I'm saying is that it is not valid logic to say that the Panasonic had CD AF and therefore all cameras with CD AF have issues with shutter lag.
As I said, my camera has CD AF and I mostly shoot sharks. I don't have any issues that I can attribute to shutter lag. I am pretty confident in saying that my buddy that was shooting an RX-100 Mk IV would say the same thing about his camera. He has gotten many REALLY nice photos of swimming sharks.
I do agree that an A6300 rig for $1100 would be a nice rig and a reasonable price. That is just up to the OP and his budget which he would prefer. The RX100 is way more pocketable than an A6300. Maybe the OP would prefer that, for use on land. Certainly not my call.
The best bet, really, is to look for a used rig here and on WetPixel. A used camera already with a housing will generally be a lot more affordable than buying a new camera and a new housing. If you give yourself a fixed budget, you can get a lot more Image Quality for the buck by going used.