Okay, it seems to be working now. It properly reflects me being a Level 2 Soldier if I click the + button on the upper right.
I think what the problem was is that I had initially built the character using the free Demo version, which does not let you advance past Level 1. After I had paid to upgrade to the full version, the UI hadn't yet fully refreshed to reflect the full version, so it still wasn't letting me add a level to my character.
So yes, my initial prediction that "it's probably a simple button that I'm not pressing" wasn't that far from the truth.
