Does permanently assigning a pilot to an island in exchange for reputation remove the pilot from your hangar?
Solution 1:
No.
It removes the pilot from your current game and prevents you keeping them at the end of the run, but the pilot will remain unlocked in your hangar (as their lvl 1 version) permanently.
Note that this only applies to the pilots with special abilities. The randomly generated pilots from the start of the run aren't kept on your hangar anyway, unless you specifically take one to the next run.