If I were you I wouldn't tell them what you're doing, and as long as it's laps just do it. Be smart about it and don't black out. A factor that could be of importance is whether you use fins or not. Fins would draw unwanted attention and could get you thrown out. The question is whether swimming underwater using the breast stroke / frog kick style, is as beneficial training as doing it with fins. But the bigger question is whether pool training is at all beneficial for spearfishing. Personally I think not.
BTW what you posted here is not new to me, I heard it before. And what they told you about someone blacking out and dying in a pool nearby is probably BS. They probably pulled that information from the internet. It's their regular excuse.
PFI, FII etc. regularly hold courses in pools. I wonder what kind of "paperwork" they provide that means anything to pool management. Maybe proof of insurance is indeed all that's required.






