Well... You could always flash
FWNitro, but you're still gonna lose Download Play, PictoChat, and your calibration capability. It just doesn't autoboot your card (it's the only full alternative firmware replacement for the DS, aside from FlashMe (which also won't help you, I just tested it on my Lite)) (So really you have no reason to do this. If you want to flash something, go carefully flash FlashMe, but it's still not gonna help you in this situation. Like I said, I tried; I put my autobooting iTouchDS RTS in my Lite (which already had FlashMe) and there's nothing you can do to not autoboot the cart)
What I WILL say though is you WILL be able to access PictoChat, Download Play, and your settings just fine if you find, via google, "firmware.nds" or "firmwaredslite.nds" and run it as a homebrew file/rom on your dstt. Unfortunately these files are fairly elusive (finding them is gonna be a bit of a challenge, but they ARE out there still) and being that it contains copyrighted code from Nintendo, we're not allowed to link it to you, nor are you allowed to ask where to download it. What this does though is boots a copy of the DS/Lite firmware, giving you full access to your main DS menu as if no card were inserted. Thus, to play your DSTT again, you'll have to power off and back on.
How did your DSTT get stuck though? Have you tried anything to get it out?