I wonder what kind of help the local, regional and national museum
organizations might offer in this situation?  Perhaps very little, but
why not try?

Contact AASLH, AAM, your state arts council or humanities council.  Ask
them about other museums like your own.  Can you approach a local
advocate for non-profits/museums/cultural activities  make it a larger
issue than personnel and micromanagement.  Remind everyone that the museum
is a public resource; public officials don't want to be accused of
shortchanging "citizens" (voters, taxpayers).

Good luck.

Dean Krimmel
public historian