The shutter lag that you're experiencing with a compact camera is caused by contrast-detection autofocus that almost all compact cameras rely on (Sony RX100 V(a) and VI are a notable exception). With CDAF, the camera processor analyzes the image output of the sensor and looks for areas with high contrast, then shifts the lens back and forth until the transitions between those areas become as sharp as possible - this takes time. DSLRs, on the other hand, use phase-detection autofocus - a dedicated secondary sensor matrix that is capable of identifying rays of light emitted by the same point on your subject and measuring how far apart they are, effectively identifying not just that the lens is out of focus, but in what direction and by how much; once this is known, the camera processor can quickly move the lens elements directly to proper position, instead of hunting back and forth for proper contrast in the image.
However, since the mirror that is used to reflect light into this secondary array covers the primary sensor, you must use the optical viewfinder in order to compose your shot. Underwater, wearing a mask, trying to shoot something in a crevice while avoiding touching the reef, this can prove challenging. Some housings accept add-on viewfinders that make it easier, but those are bulky, heavy and expensive. If you switch to live view mode in order to compose using the camera screen rather than viewfinder, then the PDAF array disengages, and most SLRs switch to using CDAF, just like a compact - with the same sluggish performance and shutter lag.
Most modern mirrorless cameras use hybrid autofocus - the PDAF points are built into the main sensor instead of a dedicated array, and are used to quickly acquire focus, with CDAF used for final small adjustments. Some, but not all, modern DSLRs have this mode as well - for example, Canon 800D and 77D can use dual-pixel AF (Canon term for on-sensor PDAF) in live view, but the cheaper 200D is limited to CDAF in this mode.
My personal experience is limited to Sony A6300 and Panasonic FX35, and while the latter had the pronounced shutter lag that compact cameras are known for, the A6300 responds instantly, and with a mirrorless camera, you can pick whether you want to use the viewfinder or rear display with no compromises in camera performance. Unlike a DSLR, you can also review the shots that you've taken in the viewfinder, since it is an electronic display, and you get focus aids such as focus peaking, magnification, and zebra stripes. Note that not all mirrorless cameras offer PDAF - all current Canon EOS M (and R), Fujifilm X and Sony Alpha cameras have it, but Olympus only offers it in their most expensive E-M1 Mark II body, and Panasonic doesn't have it at all.
Some mirrorless cameras are quite small - in particular, the Sony crop-sensor cameras (A5xxx/A6xxx) are barely larger than a compact while others use big, DSLR-style bodies (case in point: Panasonic GH5). The reduced thickness of a mirrorless camera body (allowed by the absence of a mirror box) lends itself to significantly more compact housings.
I use a Sony A6300 in a Meikon/SeaFrogs housing and while it's not a be-all end-all setup, the price/performance ratio is hard to beat.
