Callum's the one you'd need to ask as he's the one who developed the mod. I will say that it's more than likely not possible. SRB2Wii is designed under a revision of Wii Linux, a kernel to permit Linux operating systems to be booted on the Wii. So, in some cases, SRB2Wii is basically the Linux version ported to Wii.
The issue posed here is probably in the matter that SRB2Kart doesn't quite have a Linux release yet. Users have to rely on using WINE to run Kart on Linux. However, Wii Linux (AFAIK) isn't designed quite for Debian code. Along with that, SRB2Wii's kernels are revised for the sole purpose of running SRB2 at its highest potential, even on the weak Nintendo hardware.
While there is the option of using different executables for SRB2Wii, how that works is probably up to Callum. However, I'm willing to be that really no custom SRB2 clients can actually boot on SRB2Wii.
I do need to question why you'd want to play SRB2Kart on the Wii. SRB2's netcode's not corrected to work properly with opposite-endianness platforms, and on top of that, the networking's broken in the latest revision of SRB2Wii, so even if you have a LAN adapter, it won't work. It's also somewhat pointless to worry about SRB2Wii, considering its current state. The last update to the port was over a year ago, and the official client has been given some big overhauls since. You're better off just plugging your computer into an HDTV, popping in a gamepad, and go to town.
