Is that last solution possible at all? Is there a solution that I didn't saw? Recreating the mystery gift would only be possible with emulation anyway. The other problem would be losing the Mystery Gift, I doubt Nintendo will (re)release those events at all, and I would love to be able to redo the Celebi storyline or travel through the Yellow Forest again.
While I maybe could try and search for another cartridge of that generation to save my pokemons, it would probably be faster to just emulate the game, and trade with that, with the additional benefit of backing up the pokemon themselves.
Problem is, I would lose the Mystery Gift (Yellow Forest, Time Travel Celebi, and Arceus) and a lot of Pokemon I grown attached to. However, I finished the game, so there is not a lot to do, that's why I wanted to start anew. I recently dug up my old DS and Pokemon HGSS cartridge.