In my experience, and idle BROV2 draws about 8 to 10 watts (assuming you have latest R3 ESCs. Can be significantly more with older versions.) The standard battery is a bit over 250 watt hours, so that predicts about a 25hr idle life. I’ve accidentally left an BROV2 connected overnight on the bench, and it was fine. Running the motors dramatically shortens this - so depending on the loads on a typical dive for you, the impact is likely minimal. You may consider adding a solenoid or FET driven by a BR switch to your battery compartment - an easy modification that makes usage much more convenient! I’d recommend something rated for at least 100A.
I’ve also done some extensive testing of thermal stress on the internals of an BROV2. Placed sealed in SoCal sunshine, in a parking lot, internal temperatures reach a steady state at around 70 to 75C. The only performance effect measurable at the time were issues with video lag, likely due to the encoding taking place on-board the USB camera being slowed by thermal throttling. With both ESCs and general electronics, I wouldn’t start to be concerned until measured temperatures approach 80C (@ chip level.)
In warm waters, internal steady state temperatures reach about 55C, with water temperatures from 21 to 26C.