I've never been attracted to the Fender Precision. I have always liked the Jazz, but never the Precision. One day I woke up and had bad GAS for a P bass. Who knows what brings these feelings to bass players just outta nowhere.
Is there a big difference in American, Japanese, and Mexican P basses? Is the difference something you couldn't fix at a later date with some upgrades?
Are the woods the same?
My style: Finger pick only. Jazz, funk, r&b, motown, and blues. Ampeg SVT-CL amp. Flatwound strings.
American basses have a bit more attention to detail than MIMs, but in my opinion it's not that noticeable all the time. There will be lots of MIMs that are built almost as good if you look around for them. Japanese Fenders are probably somewhere in the middle between Mexican and American to same-as-American quality depending on when they were made.
An American Fender Standard does have features that are arguably "better" than what MIMs typically offer. That would be the posiflex rods that strengthen the neck, lightweight tuners to combat neck-dive, and Fender custom shop pickups. They also have a high-mass bridge and string-through body, which are upgrades for some people but not upgrades for others. The truss rod is at the heel on American basses, and on the headstock on most of the Mexican basses (some Mexican basses have a heel-access truss rod).
I say all that because you won't get a Mexican Standard Fender to be exactly like an American one just with upgrades. You can get it close but not exactly. You can't change where the truss rod access point is located, or put posiflex rods in your neck. Drilling for a string-through bridge is inadvisable.
That being said, the Mexican line is pretty great anymore. They're good quality and pretty much everything you'd need in a bass if you make sure to inspect it first, and get it set up well. American Fenders used to have better wood but now it seems people are finding both the Mexican and American Fenders are using two or three piece bodies, and neither have a real bone nut.
On Mexican Standard basses they actually use an older American Standard design for their pickups, and they use a bridge and tuners from older American Fender designs as well. You can easily spend a few hundred bucks to upgrade the bridge, tuners, and pickup to get something pretty stellar. You can still get a high mass top-loading bridge, lightweight tuners, and but custom-shop pickups. For modding, a Mexican Fender is pretty good.
If you'd like something inbetween a American standard and a Mexican standard I'd suggest the Classic 50s or Road Worn P bass, or shop around for an American Special which is about Road Worn/Classic 50s quality if not a bit more.
Anyway, don't let fanboyism turn you away from either one. Try them out. Fender does a great job of offering a good instrument at any price you can afford.
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If you are willing to get something that doesn't have Fender on the headstock I'd suggest G&L. G&L is a Leo Fender company.
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Those LB100s are incredible. You get a Plek job (makes frets very even, no fret buzz), a bone nut, and a special bridge that really makes the wood resonate. They also have really nice electronics, and lightweight tuners. The pickup is designed by Paul Gagon who worked at Fender and took apart every pickup they ever made. They're super nice. They can be had for like 900 used, but if you order a new one you can tell G&L to make it with almost any style of neck you could want. G&L is run kinda like Fender Co. was in the 60s I've heard. All custom orders, and very high quality.