Shaving the body and recessing the bridge really isn't an option, there's string ferrules under the bridge you would have to contend with as well. Cutting more of a slot into the saddles could be an option. But I agree with those saying shim the neck. That would be the easiest fix, and it would work. You could use a shim as thick as a credit card on that bass and then adjust the saddles accordingly. There's a lot of wiggle room for something like that on that bass / bridge setup to do something like that. Those saddles would then have a ton of vertical adjustment in them. It's apparently a pretty common issue even with an original neck. I just went through this issue with one myself, so I don't think your Fender neck is the issue at all. For that bridge setup I think either the neck pocket on those basses was never really deep enough, or the bridge design just wasn't all that well thought out vertically. I'd say the latter. Definitely the latter. The nut on the original neck had issues too.
I have a 1985 model that I bought for $65. It has the original maple on maple neck. It took a little work on the nut slots making them just a hair deeper and only had to to do it on the "E" string. Remember I have an original neck, not a Fender, so the nut is probably entirely different. Then a truss rod adjustment, and bridge adjustment to get it done, but it now plays like a dream.
Matter of fact it's now easier to play than my 2000 Cort Signature Series, my 2012 Ibanez SRX2EX2, and my 2018 Ibanez SR400EQM. It definitely has me rethinking the action on those.
I also replaced the electronics with an EMG GZR P set, and put D'Addario Chrome Flats on it. The sound is amazing and it cuts through the mix like nobody's business using just a Keeley Compressor and a GK amp.
I've had several guitarists / professional musicians play it along side my other basses and they all agree, it's the best sounding of the bunch. Now I wouldn't give it up for anything and I have a total of $248 invested in it, and that includes the strap and two sets of strings. I tried a set of DR Black Beauties before going to the D'Addario flats.
If you decide to forego the body(minus electronics), and the bridge, and put them up for sale, I'll buy them, like yesterday. But before you do that, I would shim it, I think you will be very happy once you do.
Here you can see the nut issue I had, even the old worn out E string wouldn't lay down in the nut.
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Both of my Ibbys, the strings lay with the top of the string almost flush with the top of the nut, I didn't go that far in making the nut deeper. But it does lay about half way in now. That, along with a truss rod and saddle adjustment is all it took.
String height before
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String height after, and I think I should drop the G a hair more, but happy with the action on it now...
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During the pup / electronics swap....
Talk about some crapped out crap. Volume was either 0 or 10, no inbetween. The tone pot was a push pull, not sure why, but it didn't work at all. Ground wire wasn't even connected. And the cord would fall out of the output jack.
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After, with thumb rest added and DR Black Beauties before Chrome Flats
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Custom Truss Rod cover from a guy out in Arizona, I splurged and spent a whopping 22 bucks.
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Pre truss rod cover and flats along side my other basses.
Left to right, of course the Epiphone, then my 2018 SR400EQM, 2012 SRX2EX2, and 2000 Cort Signature Series. The 2012 SRX is my 2nd fav playing bass and it now has Chrome Flats as well. That's a whole other story, never owned a set of flats in over 40 years until last month. Will never go back to rounds.
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I didn't spend a lot of money on the bass, and I had everything fixed with it in 2 weeks. But I did pull out a lot of hair in the process. That bridge drove me up the wall. But I absolutely love this bass now. The only issue I would complain about is weight. She's the heaviest of all my basses. But then she also has the biggest body out of the four too, so I guess that's to be expected.
I might swap out the tuners, they hold tune really well, but they have just a hair bit of slack when tuning down and then back up before they actually start to turn. It's a psychological thing, but it bothers me. I have a fear of the tuners stripping out. And eventually I'm going to take it to a luthier and get a new graphite nut put on it.
For now though I'm going to enjoy the bass for awhile.