Short version. Probably time to invest in a proper flash cart.
Longer version
While I am fairly up to date on the stuff used in straight up pirate/repro type carts I am less familiar with the multicart side of things and I doubt there are many out there that are going to be able to help -- they are not very common, not very usable and usually about as unreliable as it gets so people don't put the effort into them. Owing to the way GBA games work I can make some guesses as to the happenings but it is not going to result in a "download this"/"follow this list of steps" type thing. In all likelihood it could be reduced to a "download this DS homebrew" type thing but it is going to take a fair bit of work to get to that stage.
I assume "369-in-1" is advertising speak for "20 games but a lot of cheats/hacks for the others" or "5 games and a massive collection of emulated games". It matters little but in the case it was actually 369 games then that would change things quite a bit.
If the multicart is a slight twist on a flash cart then its original software might help, though that will also tend to want its original hardware as well which probably did not come with it. Straight pirate/repro carts moved on from rebadged flash carts some time ago but again I am not sure what would happened for this sort of thing.
After this it will likely be different to most other games, however the key word is most. Two main choices being
https://mgba.io/2015/10/20/dumping-the-undumped/ with a window or a return to the flash cart side of things, said flash cart would either be a NOR era style thing or massive RAM thing but setting pages or a more modern flash cart thing of copy to and from a NAND memory section on game selection. The linked stuff is for dumping games but the same principles apply and you will have to figure out the swap or change window command for the save section (or ROM section if you want to dump the ROMs it has).
The crazy man method would be open up the multicart, see what it has as far as save storage/handling. Dump that by something soldered onto the relevant pins, find out where the pokemon save is in it, overwrite in dump, flash modified dump back to the multicart. If you know someone of the electrical engineering persuasion then they would probably opt for this for such a small alteration.