Hey Corey! I should have revisited this post.
In the end, the problem seems to have been some combination of having the master cylinder capped, and/or not using a caliper wind-in tool. When I started over a couple days later with a loaner tool from Advantage and the cap off, it all went perfectly smooth. I finger-tightened the calipers all the way in, easy. Had to fight Bimmerlink which didn't want to turn off service mode, but then flicking the electric parking brake on manually fixed that, and everything went back to normal.
It honestly really was easy, once I had the proper tools. But of course when you follow a YouTube guide, they always miss one or two key steps.