With a Pbass, it's a lot easier to just get a lighter body than to try to remove enough material from a heavy body to get the weight down.I have a few ‘70’s P’s that I have used a new Fender Deluxe string thru bridge on, and that seems to help some.
I have a new build I’ll be tackling, a ‘79 P-Bass neck (rosewood) on a ‘78 P-Bass body, (over 8 lbs alone), and using parts I already have. A pair of vintage (80’s?) EMG pickups, going for the double P setup, and a hard to find Fender “Big Block” string through bridge:
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With this ash body weighing in at over 8 lbs, this bass could be well on its way to becoming a 12 lb beast when fully assembled. Just the body and neck with three out of the four keys installed, is just over 10 lbs… My plan is to remove as much wood from under the pickguard, minding the neck joint, leaving a bit of a boarder around the pickup rout, and attempting to leave a little tower where the middle pickguard screw goes. Going deep, leaving only about a 3/8” thickness between the cavity and back of the body where all the routing is going to be done. My hope is to keep it under 11 lbs. While still much heavier than I’d like, a slightly sub 11lb bass is something I can live with. One of my others I play rather regularly is just under 11 and it’s manageable.
That said, it will be a fun experiment. Maybe assemble it first to get a before recording to see if the wood removal makes any difference. I did a big route under the pickguard on a Squier Jag body and it didn't seem to sound different.