The whole problem was caused because the license file wasn't in the BIN.
Normally you'd get an ASP.NET error message about this, but if you'll note the page markup, there's an UpdatePanel wrapped around the upload control. This makes it so the asp.net error isn't actually rendered. I got suspicious, removed the panel, and then saw the error.
I THINK however you should check the license file BEFORE the postback, if possible, such as in the HTML module. To wait until an upload is complete isn't too good. Could actually even check it when rendering the control - then I'd not have wasted 2 days on this nonsense.
My development system has its license file in the bin directory, so when I published the website, I assumed it was moving it... ... wrong. It doesn't move those.