The solution is visit the junk shop, buy an old router (the DS is pretty picky about such things so hopefully you get one that plays nicely) that you can reasonably configure (don't know the state of ISP provided things around you, oftentimes they are limited though) and power that up/plug in the network cable when you care to play DS stuff as a secondary thing. Can go crazy with the other security if you absolutely must (mac filtering was mentioned above, as was hidden SSID even if that causes issues sometimes, you could also do the isolated subnet and what routers actually are supposed to be in the real world of networking to keep it from your other devices if that is the concern rather than your bandwidth which you could also throttle as DS is hardware limited to something like 2 megabit...).
Anyway theoretically it is possible. You would have to adapt the game to be a DSi game as baseline DS hardware has no such abilties. Decompilation is a thing on the horizon for the DS (it mostly still used C, though enough inline assembly and higher level interpreted for some things to make it tricky) but traditional assembly approaches would probably be the way. Harder things have been done by hackers (it is rather more than a mapper change but that side of the hacking world)
Once it could present as a DSi game you would then have access to the DSi aspects of the hardware which do speak WPA.
Theoretical options 2 and 3 are build some kind of GBA slot wifi adapter that can speak WPA and direct things to that (could do a DS slot affair maybe a la DS serial, enhanced flash carts or the like) or capture the data normally aimed at using nifi (local multilplayer/download play) and fire that around instead if that mode works for you. People have pondered the latter a bit more, usually wanting to play said local stuff over the internet. One of the theoretical approaches (decapitate a DS and use its guts to do the deed, speaking onwards to FPGA or something to get it into the computer, rather easier than hoping your wifi card can be tweaked to speak nifi or playing with a big boy spectrum analyser) could also be used here.
This is not a practical hack though and the benefits so very slim (though I guess with the boost from DSi mode you might gain some other potential options in this) hence not even a proof of concept.
