I tape my lips closed when using nasal pillows interfaces or nasal masks (cover the nose), but I
never tape my lips while using a full face mask (covers nose and mouth.) I'm gutsy (no pun intended!) but not quite that gutsy. When wearing my full face mask, I
want to be able to breathe through my mouth.
I've experienced aerophagia on occasions - the bloating from swallowed air can be
very painful. Very disruptive to good sleep. It's not pleasant to have "both ends" eruptions the next day either, as you mentioned!

I've had aerophagia episodes even with lips securely taped, wearing various types of masks (not a FF mask.)
I really don't think keeping the mouth shut, even with tape, would necessarily stop aerophagia from happening. You don't have to have your mouth open at all to swallow. Sometimes it can be caused by some of the air flow detouring a bit and pushing on through the wrong way into the esophagus -- even without your actually swallowing at all.
Some people do eventually get over having aerophagia, but I'm a big believer in comfort. My suggestion would be to see if your doctor would let you reduce the pressure temporarily to a level that doesn't cause you to have aerophagia. Gradually, over a period of days or weeks, the pressure could be raised back up in little steps. Perhaps given a bit of acclimatization time at lower pressures, you could work your way back up to your prescribed pressure without a return of aerophagia.
Sometimes just one or two less cm of pressure avoids the problem. One more good reason to try an autopap, in hopes that one can spend most of the night down at lower pressures anyway.
I'm not a doctor. That's just how I'd go about it.