Fix for Greasy Pulled Pork? The Problem: Recently, I've been on a pulled pork kick. My go-to method has quickly become pressure cooking, since it's relatively quick and it allows me to set the timer and forget about cooking for a while. The problem is that my meat frequently ends up being greasy, occasionally up to the point that it's nearly inedible.

The Info: I'm using bone-in boston butt roasts, between 3 and 5 pounds. I typically cut it into three or four pieces, and brown them before cooking. I pressure cook (using an 8 quart InstantPot) for 110 minutes, then let the pressure release naturally for about 20 minutes before opening the pot and shredding the meat. I shred it in the juices at the bottom of the pot (usually just a few tablespoons of broth/liquid smoke is left after cooking, along with a bit of rendered fat). 

The meat is always super tender and shreds easily, and I have no complaints as far as taste or texture. It's just so damn greasy! I can't eat more than a few bites before feeling sick. 

I'm looking for;

1.) How do I correct this BEFORE cooking? Different cut of meat? Should I trim it more? Cook it with some other ingredient? 

2.) How do I correct this AFTER cooking? Is there something I can add to either remove, or mask the greasy flavor? 

So far I've been liberally using a vinegar-based bbq sauce to cut the greasiness, and keeping my dryest red wine handy to wash the flavor out of my mouth. Can you think of any better options?
I doubt that your meat is really greasy. If it is really overly fat, you will notice it when buying it, your meat will be marbled through and through. But it is difficult to get such meat nowadays at all, since it is rather expensive to produce. 

The mouthfeel you describe is more likely to come from the gelatine. When you make pulled pork in the pressure cooker, you are converting the connective tissue to gelatine, leaving the overcooked muscle fibers swimming in it. When warm, it is really slippery, and the meat slides around in it - this is exacthly what makes for proper pulled pork that shreds. And making gelatine from collagen is not that easy. Traditionally people tried to convert as much as possible by very long cooking times, but using high pressure is much more efficient. So it is likely that you are ending up with more gelatine, or better hydrated gelatine, than with traditional cooking. 

The first thing you can try to do is to just learn to like it. There is nothing inherent in greasiness which makes people sick eating it, that's a learned aversive reaction, which can be unlearned. With the knowledge that it isn't actual fat, that reaction can subside or disappear. You can try some mindfulness meditation on some bites of the meat to get accustomed to the mouthfeel too. 

If you insist on reducing the sliperiness, you might have to make your process less efficient. This means going to a more traditional stovetop recipe without pressure. I don't know if your exact InstantPot model can cook in slowcooker mode without pressure, but if it can, you can keep the convenience of set-and-forget. You will probably have to live with other changes in texture though, not getting such a nice separation. 

Gelatine loses its gelling strength with low pH. So you are on the right way with sour sauces. You can try to marinate it with them, and keep a sour sauce in the pot while cooking it, not just adding it at the end. But then you will end with more of a braised meat, which you might not like. 

In general, it is difficult to give more advice, because this is exactly what makes your pulled pork good pulled pork. Anything you do to disrupt it will lower the quality of the texture you get.