Your analysis of the water/air behavior in your cavern sounds spot on to me.

I do not believe bentonite will seal sizeable fractures/crevices in bedrock.

Normally, we use a clay blanket with a thickness of 12" to seal a pond that is 10 feet deep. That thickness is sufficient to stop the hydrostatic pressure of 10 feet of water depth trying to go to atmosphere on low ground outside of the pond. Your situation is just the opposite of that.

How much elevation change is there from your pond relative to the highest surrounding hills? Water that enters that cave-aquifer during a big rain event could potentially have that much hydrostatic pressure trying to push out any sealing material that you put into the bedrock crevices in your pond basin.

Fortunately, that is the potential MAXIMUM pressure. Your actual required sealing pressure should be lower. You just need a seal that continues to re-route water in the aquifer past the point of restriction around your pond and send the water further down the slope of the aquifer. Unless your pond basin is the ONLY route for the cavern system (unlikely), then hopefully you can construct a seal with sufficient resistance to the hydrostatic pressure of the aquifer in your immediate vicinity.

I do not see any quick and inexpensive fixes to your problem. (If we have diagnosed it correctly.) Do you have clay available near your pond? One option might be to excavate the entire bottom of your pond again, and expose all of the bedrock. Remove all of the boulders and cobbles that can be removed to the load rating of your equipment. Pull any boulders too heavy for removal away from your bedrock exposures. Bring in clay, wet it to the optimal compaction water content, and pack in your pond bottom in 6" lifts. Maybe try a blanket of 2 feet total thickness over your exposed rock?

Sorry about your leaky pond. Hopefully you can devise some solutions!