The main issue these days isn't finding 12 (or whatever) inch LCD monitors, but finding the correct 4:3 aspect ratio LCD monitors. 12" is right in the range of POS/cash register size screens that are coming out of service, so those might be some good keywords for your search.

As for taking the curve out of the bezel, you can probably accomplish a lot before you even have a screen. Flat is flat after all. Find or make a board the right size and fill in the spaces with putty/Bondo/epoxy and get to sanding. Then you will already be ready to mount your monitor when it arrives.