Custom ROM no, however as the binary is loaded into RAM you can tickle that. Some of the wifi alt services had their hacks ported to cheat form (most of them simply remove the https and replace it with http, though owing to the way cheats work that means you need a few of them), and you could probably convert one easily enough if needs be*.
*the idea is simple enough that change https to http for the relevant addresses is the entire description, as you can not change the size you have to add something to the end and as you can shift things easily you have to rewrite the rest of the address as well as just removing https. As it is in RAM then it should not be compressed and the worst case scenario is that one of the addresses is in an overlay and you then have to do a check to see if it is loaded before doing the write which is also easy.
So yes you could chain a program called nitrohax with a DS/DS lite (DSi and 3ds will not work with nitrohax, though there may be other methods if you have a hacked one of those) and load the alt wifi hack via that which in turn could then see you do what you want.
An alternative way is dump the save off the official cart and use a ROM when you want to use wifi, or flash the official cart every time with your save from a flash cart. Frankly though I would just get a flash cart.
