Right.... there is no install for 3DNes.... nevermind about that. I really don't know why this is happening. You didn't delete the backup 3DN file yet, did you?
I know this is a stupid question too, but you're sure the backup file is renamed to the exact name of the rom and the extension was changed from .bak to .3dn, right?
Starting up the game while the 3DN was gone wouldn't have caused it. I've recovered bad 3DN files with the backup this way before. If you saved the 3DN while there was no 3DN file and your only .BAK file was there though, it may have saved a new 3DN file and saved over the good .BAK file. In that case you're out of luck.