This sounds tricky to do. Android was never meant for tablets till Honeycomb (3.*) so 2.3 tablets are running the phone oriented operating system on a larger screen, and naturally the browser was never made to give off a 'Hey, I'm a tablet' indication since the OS was intended for phones.

Yeah, its really annoying. BUT with css media queries we can take a look at device-width to make a better layout.

Just testing Bugzilla tools

Amazon Kindle Fire defaults to showing the desktop display of Wikipedia; either they've hardcoded something to fake out the user-agent for Wikipedia.org, or they're sending a user-agent that doesn't trip our mobile checks (or does trip tablet exceptions).

Are there specific tablet devices known to still have problems with the stock Android browser?

On this category there's a potentially relevant article by a usability expert:
http://www.useit.com/alertbox/kindle-fire-usability.html

Most of the 2.3 tablets I've heard of are 7-inch tablets. It may actually be better for us to keep serving the mobile site (and make it so that the Kindle Fire is included as well) for these tablets instead of trying to serve the desktop site.

We should direct them to the mobile site (and in future other tablets)
We just need to make mobile better..