How can I configure Tor, with the Flash player, for ONLY accessing region-banned videos?
This is possible. You first need to get a fake IP for the specific region, then you need to activate the Flash plugin, which by default is disabled in the Tor Browser. Check the answer of IP address in specific city for IP address and the answer of Why can't Tor Browser use Adobe Flash Player? for Flash plugin.
Stealthy is good. The other option is to use psiphon to appear to be in a different country. Also its dependent on whether the web server or the CDN/streaming server is doing the checks. I'm no expert at web development, but I imagine the html-serving servers are doing the country checks, in which case using flash should theoretically work but it will also expose your real ip address, for those who are concerned about that. Why can't someone make a gnash open source flash plugin that plays videos?!
While a lot of people are suggesting to use plugins and other such potentially risky solutions, you may want to try using a video player that supports flash. The
mpv program, for example, can play flash videos embedded within
.swf files (i.e. the same type in
.flv files). While it doesn't support a SOCKS proxy out of the box, you can use
torsocks with it to send traffic through Tor.