I have owned both a Fullerton AVRI (62) and a couple of CS P Basses (one relic 59 one NOS 62). While not owning all at the same time, all have been bought and sold the span of approx 18 months.
Firstly, I can say all of the basses were exceptional in terms of sound/tone and build quality and I think you’d be happy with either of your options. Ideally if you go down the Fullerton path, I’d recommend seeing/playing first, as there are some quirks such as ski-jumps to look out for, as well as lots of non-original pieces out there. In my opinion, if it’s not all original, it’s not worth the expense.
My current p bass is a early 2000s NOS 62. For me, it has been the full package. The fit, finish and tone are all exceptional. It feels and sounds like it’s price tag. While some opinions are that the NOS models are in a similar league to the latest AVRI/American Original Series…having also owned a NOS 64 Jazz Bass which I did have the chance to A/B with an American Original 60s Jazz - I stand by my thoughts that the NOS CS models are in their own league.
The relic 59, also sounded fantastic and was still obviously a high-end bass, I personally just could not gel with the relic’d neck, nor the aesthetics. Each to their own.
The Fullerton had loads of mojo and sounded maybe the best of the bunch by an extra 5%. I sold this bass due to covid last year and while I sometimes wish I could have found another when I was in a position to replace it, the NOS 62 is ultimately every bit as good (give or take that 5% of tonal difference) and by comparison does “feel” like a $4000 bass. Whereas I felt with the fullerton I was paying a premium for that MOJO and it did not “feel” as expensive.
I’d say if you have the chance to play both, that would be ideal as you can make the most informed decision. If you have to buy online/remotely I’d suggest there is “some” risk involved in the fullerton. Where as from my experience, a plus for the custom shop is that there is an element of consistency there, having owned 3 including the J bass - all of which are/were exceptional basses.
All the best with your decision making and search.