I have a bathroom in my basement with a flush-up pump (the exit
Plumbing is above the basement). It has worked fine for several years
But is now starting to stink. I think what is happening is when the
Toilet is flushed the water in the toilet completely exits
Momentarily allowing tank gases to escape in the room. The way the
Water acts in the toilet leads me to think that there is an air tight
Connection between the toilet and tank.

First, I'd like to minimize the tank odor by neutralizing it somehow.

Second, I'd like to change something so the water doesn't totally
Evacuate from the toilet so a seal remains.

Any help would sure be appreciated.

Check your vent piping.

Doug gives good advice. There should be two vents, one venting the pit and another venting the fixtures. Check them both. Good luck, Tom