How should I choose a BBQ sauce for root-beer-braised pulled pork?

A Southerner will probably tell you that there is no type of BBQ sauce to go with this type of pulled pork because it's not actually real pulled pork... or real BBQ. Authentic pulled pork is smoked, and that comes with a completely different set of pairings. This is technically just braised, shredded pork.

Which is not meant as a criticism, mind you - I often make a similar recipe in the winter time using a dutch oven. But terminology is important here because there's a huge difference in flavour between pulled pork and braised pork. I'll note for the record that any kind of slow-cooking will make a pork shoulder fall right apart; you do not need the root beer (which isn't that acidic anyway, and acid-tenderizing is somewhat of a myth) to tenderize the pork. The heat and moisture do that all by themselves, although root beer is as good a choice as any if you're braising.

Now onto the flavouring. Because the braise, unlike smoke, only minimally flavours the meat, you're going to need strong flavours from elsewhere. If you're really trying to mimic an authentic Southern-style pulled pork then you need to incorporate the smoky flavour somehow, which means using a smoky BBQ sauce (hickory is common) or better yet, incorporate this flavour at the very beginning of the cooking process by applying a dry rub containing smoked paprika or including a small amount of liquid smoke in the braise.

If you do use a rub then the other flavours are up to you - there are plenty of recipes out there - just don't use salt in it because pork already has plenty. As with any braise, stick to aromatics like garlic/onion powder, various chili powders/peppers, cinnamon, etc.

Once you've successfully incorporated the smoke (or what passes for smoke in indoor cooking) then a little sauce goes a long way. Definitely incorporate some of the leftover braising liquid so as not to waste those delicious drippings - I use about half of it but that's with a much milder braise. Ketchup and dijon mustard in a 50/50 to 60/40 ratio is very simple and surprisingly effective when mixed in with the juices.

If you still find this to be lacking punch, you can throw in some cider vinegar, which pairs well with both the pork shoulder and your root beer braise. I actually boil off the pan (after reserving some braising liquid) and use cider vinegar as a deglazer for the fond, but since you're using a crock pot (or equivalent), you probably won't have much/any fond and this would require an extra vessel, so... just throw a splash or two into the finished sauce, and reduce if that makes it a bit watery. Worcestershire sauce is also a good way to punch up any homemade sauce, and if it starts to get too intense or tangy from either that or the vinegar, tone it down with some brown sugar or molasses if you have any.

If you're dead-set on following the recipe to the letter, then, as above, I think I'd choose a hickory or mesquite sauce, something smoky and not overly thick. Some of the better brands/products will actually include a small amount of alcohol, like whisky or bourbon. Almost all will contain onion, garlic, paprika, sugar, molasses, and vinegar (often cider vinegar, as above) - not surprisingly, exactly the same kinds of ingredients as I'm suggesting in the dry rub and homemade sauce.

There are as many sauces used with pork as there are states below the Mason Dixon line. And most of them serve sauce on the side. When I make pulled pork, I smoke it with nothing but a dry rub, add a little vinegar-based finishing sauce after I pull it, and have several sauces on the side to suit varied preferences among my guests.

I'd recommend a minor departure from the recipe. Use a spice rub for cooking, one that includes salt, pepper, chili powder, cumin, onion powder, and garlic powder. Add a splash of finishing sauce after you pull it -- vinegar, garlic powder, a squeeze of mustard, and a dash of hot sauce. Then pick a couple of different sauces to serve on the side.