Update/resolution: It took some contemplating but I figured out the issue and chose an alternative route to solving it versus the most correct method, which would have been to drop the tank and make lots more work for myself.
The issue: The vent pipe was too long. Whomever installed it appears to have put it together backwards in my view. The top of the vent pipe stuck out of the roof about 1.5", so it sat inside the vent cap just fine. However, the bottom of the pipe was about 2" from the bottom of the black tank. Based on the amount of effort it took to turn and move the pipe, I do not believe it slipped down at all. I believe it was installed that way. Another clue is that you'll see in the photo, there is an adapter glued onto the end of the pipe inside the tank. I believe this is the piece that should have sat on top of the tank so that the end would be through the seal and slightly below the top of the tank. As it is, no amount of force could pull that adapter up through the seal without pulling out the seal itself and requiring the tank to be dropped. Note also how far down into the tank the bottom of the pipe is after pulling the adapter up to the seal. There's no way it make sense to have that adapter on the inside of the tank.
The solution: I pulled the pipe up as far as I could without causing damage to the seal. This required all the force I could muster. I removed the toilet, floor flange and flange seal so that I had as much access as possible into the tank. I then used a 12" drill bit and 90 degree drill to drill holes in the pipe just under the seal at the top of the tank. The drill bit was just long enough to reach. It wasn't a pleasant job but I flushed the tank and used a garden hose with sprayer to spray all sides and top of the tank before I started. I also had to cut 2.25" from the top of the pipe on the roof so that it would fit inside the cap, and I drilled holes in the pipe so I could push a metal dowel through which rests on the roof to hold the pipe just in case it would slip in the future. Following are the pics...
This pic shows how I used a pipe inside the vent pipe so that when I turned and pulled on the vent pipe, it would not collapse.
This pic poorly shows the vent pipe after I pulled it up and inserted the metal dowel to hold it there, although I am not sure it would have slipped anyway. The dowel has a bend on one end so it doesn't appear to be straight through the pipe even though it is. This is also before I cut off 2.25" from the top of the pipe, then reinstalled the vent cap. I also caulked around the pipe itself prior to installing the cap assembly.
This pic shows the vent pipe/adapter inside the tank after I pulled it up as high as I could. The top of the adapter is tight to the seal. I then drilled a hole, went to the roof to turn the pipe slightly, drilled another hole, repeat, repeat, repeat. For some reason this pic posts upside down...
We tested at each step to confirm any improvements. Before starting this project, the toilet would begin burping at 30%. After pulling the pipe up so the adapter was butted up to the seal, it would begin burping at 50%. After all was said and done, I could fill it to 100% (water up to the bottom of the toilet flange) and it did not burp at all.
I hope this is useful for someone out there. At minimum it may help some understand their system a little better.