Why should some repair shop delete your homebrew/cfw installation? With what authorization?! Replacing the gamecart reader has no impact on the installed software. I would simply forbid them to modify the software.
On old 3DS and on both 2DS models this requires tiny soldering. This means whoever attempts this repair should be skilled with the soldering iron.
Backup your saves with JKSM and/or Checkpoint in decrypted form, because a 1:1 copy of the SD is not helpful if the console is bricked (this has nothing to do with your planned repair, any device can randomly fail). Of course the complete SD backup makes sense, too and is usable as long as the console works.
