While I haven't used it myself, it is my understanding that RX100 II, like most compact cameras, suffers from significant shutter lag due to only having contrast-detection autofocus. RX100 V is the first model in the series to add phase-detection AF, but it is significantly more expensive, although the rumored imminent introduction of RX100 VI might push the prices down a little.
PD AF is, generally speaking (but not always), better. But, I don't get the impression that the OP is looking to drop $2 - 3K for a camera rig and I think that is what you'd be looking at for a camera with PD AF plus a housing. And, of course, it's kind of silly to spend that much on a camera and housing and not spend for the strobes you would really need to get the full benefit of the camera.
The RX-100, pre-Mark V, can take FANTASTIC photos. I have dived with a buddy who shot some award-winning photos with his. Actually, I think just about any halfway decent camera can get really good photos if you have good lighting (usually meaning good strobes) and the skill to capture the image.
Also, even my "fancy" Olympus mirrorless camera uses CD AD. It is also capable of capturing very nice images. A camera with good CD AF is still good. Probably even better than a "nicer" camera with a poor implementation of PD AF (a la some early 4/3 format cameras, I think - not the newer micro4/3).
In short, the suggestion that an RX-100 Mk 2, Mk 3, or Mk 4, is anything less than a really nice camera for u/w shooting is, in my opinion, off base.
I shoot with a Meikon housing. They are really inexpensive, but mine works well and the few reviews I've found here or on WetPixel.com all seem positive. The only time I've ever had any issue with my housing (in over a year with it now), is diving the Tarpon, off of Hatteras. I was at 135 feet or so and some of the buttons were being pressed and held down by the water pressure. But, the housing is only rated for 132', so I'm not too upset about it. At least it has never leaked!
If you want to judge for yourself whether CD AF is any good, I posted some photos from my outing in June here:
Wreck Shark Shootout 2017
I feel like it's only my skill, not my camera's AF system that is holding me back.
Like I said before, an RX-100 Mk 2 with a Meikon housing, for around $750 total, is a really hard package to beat for someone who wants to try u/w photography but it not yet ready to drop the big bucks.
Used is also a legit consideration. I only have my Olympus mirrorless camera because I found it on Craigslist for $400 with the kit lens plus a decent longer lens...