You could try going to city hall in person which makes you look more invested over a phone call. Try calling a civil lawyer because they understand legal matters. There’s probably not much you can do besides these two things.
Yes, a writ of mandamus may be appropriate here. The OP should consult a local attorney. Sometimes, things like this can get resolved with a phone call from your attorney to the city attorney. Other times, you have to litigate it.
You might call the City Attorney and tell him/her you’re seeking counsel to get that writ thingy.
Might be just the push the City needs to come up with the permit.