Feedback can be caused by gain-staging and volume, too. How much gain are you hitting the amp with?
If it were me, I might just return/exchange within the 21-day. Bummer because it takes time for shipping the exchange.
FWIW there are things you can do without returning it that might solve it, like putting 2-sided tape on the slug bobbin under the cover, or a little bit of wax on the slug bobbin and melting it into the cover with a hair dryer (not too hot, but enough to keep the cover/pickup from oscillating under high gain. I honestly haven't had to do these myself, I've just read repeatedly on here and other forums how guys have done that and it solved their problem.
I'm gonna go wayyyyyyyyyy outside the box here. Go back to whatever pickup of the first three you like the best and try a really good parametric EQ.
This almost sounds like it is a structural issue with the guitar. Like an acoustic that just has resonant frequency that feeds back. And a really narrow notch filter (AKA super tight parametric EQ) would be my solution.
You seem to have more than enough pickups that almost do it for you.
Look into an Empress Para EQ Deluxe, or something along those lines. I suspect Revival has something like that as well.
I actually have a Wampler Equator in the mail, but I'm planning on putting it in the effects loop. If it doesn't work there the way I'm hoping, I could move it to the front of the amp.
Last night after band practice I pulled the 59/custom hybrid out and put a Custom 5 back in. The uncontrolled squeals are a little too much. I'm playing in a different room now, with some sound treatment, compared to the last time I tried the custom 5. I'm hoping all the bad junk I was hearing was from the room. I'm about to go practice for a bit and find out.
PAF Pro ?
The PB is a good choice. It can get really thick with the bass side up too high so play with the pickup height to get the balance you like with your rig. I love the PB too and have it in a HSS Strat with 2 STK-4 Classic Stack Plus singles. It's a great setup for anything.