I think your suspicion that the bass is worth more as parts is entirely accurate. Selling that as a put together bass will get you some money. But whenever you part them out you get a bit more, as a guy who really needs a 63 P neck is not going to pay more just to have some Jazz parts he doesn't need.
Your P neck is looking good, as it has the original tuners and finish. The chipped decal is actually more of a help for you, as it is fairly authentic. It should get some good money on ebay. Not as much as if you had sold it 2 to 3 years ago, but that is life.
The body is a beat bad refin. A really bad refin. Hopefully the neck plate matches the date codes on the pots (and they are original) and the pups may have a date written on the bottom unless they were swapped out. It would get some interest on ebay, but with the bad refin in the neck pocket and bridge missing, I would say you have to take the rest of it apart to verify originality and include them with the body if you sell. And throw the black strap locks away.
The body guard is a curiosity. Lots of people pay ok money for them but nobody really loves them, and it isn't hard to find them in the box never used. Keep it with the body, but keep in mind they don't belong original to any Fender Jazz.
OR, you could go to the trouble of refinishing the body, and buying parts that are missing. A lot of work, a lot of money, and you could easily pay for a good original 62 reissue Fender bass that would always hold its value. And would play great. I personally wouldn't bother. Sticking a Jazz neck on it with a new finish just makes it a refinned non original Jazz, and an original bridge will set you back a lot of money. Oddly enough, 66-early 68 Jazz necks with the old decal and binding/blocks have more demand than older necks now, so you would be in competition. You could try a simillar route with the P neck, and find a loaded 62 reissue body on ebay for maybe around $500, but you will always have a real neck on a body that is not original, and that is not worth much more than just the neck.
Or you could try to work a good trade for the parts to end up with one or the other in some sense of being more what they should. That could take years and someone will try to milk some money in the process.
I'd sell it and get an older reissue Jazz and maybe have a little left over.