If you are on crunch time, that makes it a tough situation.
It's a long shot IMO, but can you get a part number from the driver? Maybe there's an on-board way to adjust it while you have it apart looking...if it's not obvious, a part number may help locate a manual.
Also, if you limit your search on eBay to USA sellers, there are some remotes as well as really cheap remote+RGB bulb combos that might work for you. (Just don't use the bulb part of the combo.)
Again this is not from experience, but my understanding is that these RGB remotes can be generally interchangeable....and these seem available from US sellers from around $5.
Depending on your willingness to drive for a same-day pickup, I even found some on eBay "close" to Austin. Do an Advanced Search and limit by town or zip code if you wanna take a look.
A few well-placed questions to confirm compatibility (if your current vendor can't confirm for you) and it seems like you may have some options well short of waiting another month for the next China-boat.
FWIW, this lag (hidden cost!) is definitely something you quickly learn to either plan for or avoid. There's a reason the prices are cheap...nothing comes for free. :-)
Though I haven't had any big problems with missing or broken items so far (*knocking on wood*), when I'm ordering GU10 bulbs and supplies on eBay for my projects I usually order 10-20% more than I need of any given part. Nothing in life is perfect, so the extras usually pay off in multiple ways.
(Still doesn't guard against a case like yours where the vendor leaves something off an order....)