That large cup sized drill bit that bores holes for the lock set can be used to fix your hole problem. You will need a 3/8 drill and that special drill for lock and handle holes.
Find a trash door, or even use a 1 x 6 and drill out a hole as you normally would for a handle. Then when drilled, carefully remove the plug that you cut out. If you are using a 1x6 then drill two holes.
Now, with some glue and some finegleing, you can insert this plug (s) into the old hole that you want to seal and your set. AFter it dries, then sand it down even, might have to use some drywall compound to even it out. after done, sand again, prime and paint. You might never know there was a hole there.