After nearly a year, I finally sanded down the patch and tried to put up the towel bar again. Part of the problem is where the studs are located. Without putting the bar behind the door, we either need a really short bar or be willing to anchor a one side in the drywall. We opted for the latter.
My first attempt tried to anchor the bar at the edge of the patch. Needless to say, it didn't work and I had to put some more plaster up on the wall. For the second attempt, I decided to go inches above the patch site, and thus also almost out of my youngest son's reach. To my surprise, not only did it work, but it ended up being the same height as a towel bar on the opposite wall, almost looking like I had intended it that way.
I know that it was only a minor accomplishment, but it still felt nice to get something done.