Hi TCB
Welcome to TFP!
Since a permit isnt required, I definitely agree that the pool be built to code, whether a permit is applicable or not. This should be written into the contract with the PB. Dont accept a gentlemens agreement on it.
Plumbing doesnt usually require an inspection unless it ties into the city sewer or water. However, it should be pressure tested to 30 psi over an 8 hour period. If there are any tiny leaks or weak joints. The pressure test will confirm it.
Definitely dont accept any electrical work performed from anyone less than a licensed electrican with pool experience. A good electrician will know the pool building code and will do it right. This includes the bonding of the pool.
If you have a main drain in the pool, the main drain must conform to the Virginia Graeme Baker Act. Its a safety consideration so someone doesnt accidentally get stuck on the drain due to the suction of it. No PB in his right mind would skip on that. But if you have a main drain, just make sure the PB knows it needs to be coincident with the VGBA. FWIW, its not even necessary to have a main drain in a swimming pool.
As to whether or not you need an independent inspector is a tough question. There are a lot of PB's who do excellent work and dont cut corners, and then there are 'others'.
I would recommend talking to the PB about how to reconcile the work performed against the building code and see what you can come up with.
PBs generally know all of the inspectors and may be able to make a recommendation to perhaps a retiree or one who wouldnt mind some off duty work. Hard to say what the case may be, but I would recommend to maybe thing about something like that.