Hm, that's an interesting bug. I'll try that today or this weekend with my Belkin keyboard, but I suspect I'll have the same trouble. A quick googling of the issue demonstrated that this issue has been seen since at least back in 2003 on Windows Mobile OSes. I'm wondering if the TS client uses a different keyboard wrapper to communicate with the remote server, so that the wireless keyboard receiver app on the Axim is simply passing on instructions to the "regular" keyboard tool that might suspended while the TS client is active/in focus.
I might also try seeing if it matters what keyboard app I use, either my Calligrapher 8.x keyboard app or the regular "built in" app. I'm guessing it won't but it's worth a shot. Just for kicks I may try using my
Pocket Controller app with the Axim docked and see if a similar behavior is observed when using my desktop keyboard.