Ah the craft of the Luthier. For what it is worth I had a bass guitar that this reminds me of. My old epi. It was an old epiphone p style bass bought under a dent and scratch policy. Nothing wrong with it except it did not say epophone, and just said epi. Thing had a nightmare bridge that kept causing stuff to slip. I eventually took the guitar apart and installed a new bridge and new strings so it was and intune.
However like you said the guitar is indestructible to the way it was earlier, which while annoying you could also work around. The easiest way to do this is to have the character take out the strings which if the strings are indestructible, well he might have to be careful. I remember one time I took a string out wrong and the thing whipped out and hit me in the face leaving a nasty little cut. Six indestructible coiled steel strings? Could be one hell of a nasty whip if you ask me. He could then uninstall the bridge, which he could do a lot easier. Well he could if it were an electric. In this case I would say that you are in a bit of trouble.
Other things it could be besides a club since it is indestructible
A really nice shield
An amazing battering ram
A pretty useful brace
A highly illegal cricket bat
There are other things I am sure but its too early for my Macgyver skills to be functional.