Eepinator should be able to write a save from any flashcart that has DLDI support to a commercial DS game cart. The wifi apps are only needed to get a save from a commercial game cart with certain slot 1 flashcarts.
Eepinator works just fine with slot 1 carts, but in order to get the save from a retail cart to a slot 1 cart you have to swap the flashcart out to read the save from the retail game, then swap it back in to write the save file to the memory card.
Some flashcarts (like the R4) will not allow access to the filesystem if they are swapped in while the system is on, while others (like the DS-X) will reset the system if they are inserted while the power is turned on. Other flashcarts will just work.
That doesn't apply to this question though, because you don't need to swap the flashcart back into the DS to transfer a save to a retail game.