I don't have the technical knowledge to know if it's even possible. I'm merely hypothesizing based on the fact that some degree of TWL_FIRM patching is possible. The overall solution might likely be 1-3 patches put together between TWL_FIRM and homebrew software. For the purpose of this post, a "patch" is referring to a patch to TWL_FIRM or an update of the Wood R4 Kernel (or equivalent DS flashcard loader).what kind of patch?
The first patch would be finding a way to reinitialize the SD slot while the console is in DSi-mode. This may already be possible though. Watching the video of ApacheThunder's DSi homebrew launcher makes it seem like that's already possible.
The second patch would be to redirect the filesystem calls that Wood R4 Kernel makes to the SD slot. I foresee an obstacle being the availability of source code for the launcher. This may very well require REing Wood R4 to make it work, or it may take creating a new loader from scratch.
The third patch would be to handle gamecard emulation. This is probably the biggest hurdle of them all, although looking at a DS emulator's code may be useful in creating such a patch. Since my technical knowledge of how the 3DS operates is very limited, I don't know the feasibility of such an endeavor. If DSi-mode underclocks the 3DS processors, couldn't the unused power of the processors be tapped into to handle the emulation? This would likely need to be applied to the TWL bootloader and possibly be an obstacle itself.