You can delete the save in-game, (UP, Select and B at title screen), but the thing is the Mystery Gift option isn't officially unlocked until you reach Jubilife and talk to a NPC.
Though, you probably can still inject wonder cards, but only after a character gender and name is chosen, and game is saved.
Sadly you'll have to do this. I've tried injecting a wonder card into a blank save, then start a new game, but the game wipes it.
Same for Gen 4. If I'm not mistaken, as long as it's not received from a deliveryman, the correct event flag doesn't get set.