Welcome to ceramic pickups. The Monster Tone P pickups you have are ceramic, which can run hot compared to alnico. I think lowering them will help, or adjust the volume on the bass or the gain. I have the same thing going on with my DiMarzio ceramic pickups. They sound really good though.
The lack of a cover is okay. 50s P bass pickups all are without covers. It’s the way they were made to begin with.
I usually prefer good, bright ceramic pickups over alnico. They just fit with the styles I like. I’ll take punchy and mids-focused over softer and warmer on most days.
They get a bad rap because they are cheaper to make and thus a lot of cheap bad pups also tend to be ceramic. But that’s because they happen to be cheap bad pups, not because of the ceramic. I like the ceramic tonal quality better, and high quality ceramic pickups like DiMarzio and SD just sound better to me than alnico. YMMV.
The pickups cannot go any lower. I feel sorry for the foam under them.
In all the research I did before buying this bass, I did not notice that. Or I did and didn’t realize what it meant.
Don’t get me wrong, they sound really good. I’m just surprised at how much hotter they are. It’s very noticeable between the P-4 and the B40, both passive basses. When my SDs arrive I’ll put them in, and if it turns out I like the MonsterTone pups better, I’ll put them back.
Ceramic and alnico magnets can both sound excellent or terrible. The magnet material is just one parameter of the pickup, the goal being to match all parameters to get something interesting at the end.
Actually the split coil design with cover is a '57 design but yeah, early P pickups did not have any cover, which is pretty bad because the magnets are effectively what ties the pickups parts all together (see the picture : a '57 type half-pickup without its coil). It can be very fragile if you use the pickup as a thumb rest. This issue does not exist with covers, or with a pickup frame made out of one single piece of plastic.
If this is particular bass is hotter or opposite weaker than the rest of your basses a decent DI box can usually fix that problem. It converts hot instrument level to an easier to live with line level. Here’s a better explanation video.
It’s easier the opposite direction, if your pickups are too weak you can just get a booster pedal and it should solve your problem.