The only way to seal it is to stopper it from the tub side.
There are usually slots at the bottom of the plate that drain the water. There is a product that I think is called Dux Seal, which is similar to Play Doh, that is sometimes used by electricians to seal the end of conduits. You could put some of that over the slots, and the remaining area around the drain actuator would not leak much water.
Duct tape would probably also cover the slots if you clean the area well first.
You could put a piece of Saran Wrap over the whole thing and the water would probably hold it in place after the water rises over it.
You might take the overflow mechanism off the tub and try to fit a rubber gasket behind it, with a hole for the drain actuator, to effectively seal the overflow.
You like a full tub, but you must realize that your are creating a flooding risk if you were ever to get distracted or called away to answer the phone while the tub fills.
These are ideas and have not been tested. You could probably find a local engineer to help with some testing.