No, you can't save them once their in boxes.
If you have a friend that plays Animal Crossing or a second town, you can get Lucky to move there by having their characters talk to him. Then once you complete the sixteen villager cycle you can move him back. However, I don't recommend this as it takes a while and there's always a chance he'll move out of the other town.
I think your best bet is to just let him go and get a new Lucky.
Sorry, bro.