Looks like Beaker might "win the prize". This morning the problem went away. Recap - we started this trip five days ago with clean empty tanks, but suspecting trouble ahead. Each morning I would dump 4 oz of RV toilet treatment into my 38 gal black tank. Around day 2 1/2 the burping began. Last night the burping stopped and hasn't come back. The tank is two-thirds to 3/4 full.
This rules out my too-long vent pipe theory. But if things are backing up in the elbows, as I now suspect, that might also partially explain the leaky toilet base, although I probably still need a new seal.
I do have a clear collar so I will be watching close for more clues when we dump and leave this afternoon - I'm not in there every minute - but I think the problem has been found. Thanks again everyone and know that I will be rinsing well.