Very interesting. I tested with my DS Phat, slot-1 Supercard DS One:i, using the English version of Super Mario Kart. I was able to see the title screen and the demo that plays if you don't press any buttons, but was not able to get in-game. The entire DS froze when it should have showed the character select screen, and the buttons and touchscreen stopped working, so I had to press the Power button to try again.
I was using "SNEmulDS_dsp.nds" and the "snemul.cfg" file from your link, with my roms in /SNES/
Should I try using the PAL version of Super Mario Kart? Of course I understand this branch is experimental; just hope I can get as far in-game as you have.
Congrats on the progress! I never thought I'd see this day.
EDIT: if anyone is interested in a blast-from-the-past, here's a very old crummy Youtube video I once made that showed how Super Mario Kart worked (it didn't) on Snezziboy for the GBA, which eventually turned into SnezziDS for the DS. The context behind the video isn't important nowadays. Just watch from this specific timestamp (5:36) onwards: