Franken P Blues..Oh Man!?

It appears the saddles are bottomed out so there are two ways to get the action down. Shim the neck or rout the bridge an 1/8" into the body.

 

I certainly know that is an Epiphone body. Not sure if it's an Accu Bass or some other model. I say that's the main problem. Epiphone makes some crappy basses. Although it is, in fact, a P bass type body, I'm not quite sure it goes well with a Fender neck. Maybe the heel of the original neck was somewhat thicker than that nice Fender neck you have now.
Option 1 (easy/cheap): Get a shim from Stewmac. That should fix your high-action problem.
Option 2 (not tha cheap): Get rid of that Epiphone body. Get a new (Squier maybe) P bass body that goes well with your nice Fender neck. EDIT: that should fix your heavy-weight problem.
You have some quality parts there! Nice neck, nice tuners, great pickup, new electronics. My equation still points to the crappy body.
Either way, believe me, you won't get your money back. But why not make it fun? I love parts basses. Had an idea for a parts bass for 8 years and when it finally got done I was super happy with the result. Both my basses are parts basses and I don't need anything more (for now).

If you ask me, I'd go with option 2

 

I agree with above post, get a new (Fender, Squier, or licensed) body and bridge to go with the neck and other great parts you already have. You won't get lower action with that bridge, and it doesn't appear to have standard mounting to make a straightforward swap for a regular bridge.

I found a loaded Classic Vibe P body on eBay for $140 and am building it with an MIM Fender neck. Should be pretty easy to find something like that.

 

Try a vintage bent plate fender bridge and see how that works.

Shim neck if needed.

 

There are several approaches for the high action. It appears the saddles can be lowered a bit, the saddle grooves can be deepened and the neck can be shimmed. For me inletting the bridge into the body would be more work than necessary.

The bridge can be replaced with a Fender style threaded saddle bridge or similar with thinner saddles which may help.

I wouldn't rush to getting a new body until those options were exhausted. The first are free to the DIYer and the new bridge can be used on a new body if it is necessary to go that route.

 

If weight is a serious concern then get a new body or part the whole thing out and pickup a different P.

If not a neck shim (you can get the fancy full-contact ones) should make it so that the bridge saddles can be raised to properly set the action.

 

While you’re waiting for the fancy full contact shim, you can fake it with a guitar pick or a piece of an old credit card between the neck and the pocket, at the end closest to the bridge.

 

This is totally accurate. There is much conjecture as to whether or not a full contact shim is really superior to traditional ways of doing it (Fender used thin cardboard).

 

I suggest drilling holes in it.

 

If you like the neck and the tone of the pickup, sure, maybe a new body would work, but there's no way of knowing until you try it. Looks like you'll need either a string-through body with ferrules that line up to those bridge holes, or a new bridge with holes in the back if the new body isn't drilled. If you don't like it, then you're $650 into a bass you don't want to play.

I'd personally part it out and get a different bass, there are some fantastic P-basses out there - I got a phenomenal used American G&L SB-1 here for less than that recently.

 

Why are people saying the saddles are bottomed out? Looking at the photo you can clearly see the underside of the saddles reflecting off the base plate. Looks to me like those saddles can be lowered at least 1/8". Or is the photo just tricking my eyes? I don't see any reason you couldn't have that bass playing decently with about 20 minutes of setup. A shim is super easy to fabricate and install if it needs one. But it'll still be a heavy beast. I'm near Oakland and I'd be happy to help you get it setup if you want to stop on your way to Sacto.

 

Don't chuck the body. Have fun with it. Strip the top and sand about 3/32 off the top. The bridge is out of alignment, reset that. Flat black would look great with those black block inlays. Do it yourself and feel good about it.

 

How about getting an estimate from a well known setup guy. You should be able to run through all options and cost.
I realize you’ll do the work yourself. This may save time and money on parts not needed.

 

I had one of those bridges on a Fender-ish parts bass and it always struck me as too high. I messed around with shims and considered modifying the bridge saddles at the shop, but in the end, I just swapped it out and everything was easier.

Also: I wouldn’t be surprised if that body was made of plywood if it’s that heavy.