Random permutation of bit strings of length $n$, where $n$ can be any positive integer?
Format Preserving Encryption is the name for a standard way of doing what you are asking.  It does a keyed mapping from a (potentially small) domain to itself.  It does more than you are asking (it tries to be secure; you specified that you didn't need that), however there should not be an issue with this.

You have already found it yourself, however if you were to ask me which FPE primitive to use, I would have suggested FF1.

(I'm putting this as an answer so you can accept it)