Is it legal for the building inspector to not close out a trades permit because another part of the job wasn’t permitted?

Similar examples of illegal discrimination would be "Tesla employees only" or "no college students".Therefore, going back to your original question, I think you would perform background checks and employment verifications as usual.

Did you not pull the permits when you were renovating?

I just bought a live-in flip in College Grove, TN (closing on the 12th).

You can also consider renting room by room, think of things like this near college towns. 

Look to see if any permitted work has been done. 

If mom and dad pay the bills while their 19-year-old lives with them and goes to college, I still run a credit/criminal on the kid to ensure he's not a drug dealer, sex offender, or whatever.

I actually chatted with the local owner and he said he would've put the money in but his son is not interested in his business being a college professor in a foreign country working in the field he enjoys.

I am in the process of obtaining a permit for residential/farm crossing and am waiting to hear back from them.

I looked up the property, and it's zoned for high-density residential, and it's three blocks away from a regional university (I live in a small college town).