However, there is a model number on the TV. And after a lot of searching on Google, I believe I have finally found an original equipment matching remote (as opposed to the DirecTV remote that one seller was pushing as a remote for the set -- I have one of those and it almost does some of what I need correctly, but mostly it controls volume) that should be coming to my door for less than $27 shipped.
So I'll keep my fingers crossed. :)