David
A couple of thoughts from a lifeguard and five-day-a-week pool user who practices breath hold, underwater hockey, and scuba skills in a pool.
I think your chances are about 50-50.
If they object, it may be on the grounds that the use is incompatible with open swim, people going off the diving boards and landing on divers and so on. Does the pool have a prohibition on swimming with mask, fins, and snorkel during open swim? Many do. If so, they will by extension, oppose scuba during that time. If not, point out that putting on a tank is just like snorkeling, etc.
The municipal pools in Anchorage until recently had "scuba nights" open to people who wanted to snorkel and dive. It was a way of filling the pool with paying customers when the pool would otherwise largely be empty. If the aquatics center is in a budgetary fight for its life (as many are across the country) they may be amenable to having this as a "new" public use as a way to expand their clientel, especially if it happens when the pool is underutilized.
You mention that they already teach scuba classes there. I would venture that the dive shop is actually renting the pool during that time. If so and you can fit it into your schedule, contact the dive shop owner and negotiate a reasonable price for you to practice while they have the pool. I know two shops up here that will allow that arrangement for a very reasonable cost ($5.00) which is less than the normal one time entry for open swim. If the county runs the lessons, ask if you can hop in during that time.
Private health clubs with pools may be a little more lenient than public pools.
If all else fails, figure out how to hop the fence after hours.