Don't know if you'll like this idea but I've had fun thinking about it:
Treat the beer mix as a brine but reserve a cup or so before dumping on the butt. After soaking, rinse the butt, dry it, and apply your rub (you might need to decrease the salt in the rub depending on how much salt is in the brine).
For a sauce:
Slice 3 or 4 large onions and carmelize in unsalted butter till nicely browned. Add 2 clovees minced garlic, cook till fragrant but not brown, about 30 secs. Add a tsp of dry thyme. Push the onions to the side of the pan an add 1/4 c or so of apple juice, a tblsp of mustard, and half of the brine you reserved earlier. Reduce till syrupy then puree and return to pan. Adjust seasoning by adding some of your rub. Adjust comsistency by reducing further or by adding more of your reserved brine.