CV P-Bass Question #2 (electronics upgrade)

After repairing the wonky input jacks on my kid's bass amp tonight, I noticed the volume pot on his CV P-Bass was scratchy. I coulda sworn I had replaced all the electronics in this bass, but discovered when I opened it up that I had only replaced the jack... it still has the original cheapo mini-pots. I sprayed them with cleaner and the scratchiness cleared up, but I need to replace it all with good stuff.

I checked the Seymour Duncan website, they recommend 250K pots and a .05 mfd cap. Are these pretty much the usual values for a P-Bass?

I noticed two things about the existing setup: the volume pot does all its "turning down" in a very short range. Only a tiny bit or the top of the rotation is useful. Also, the tone control does almost nothing: at first I thought is was open-circuited, but if I listen really carefully, I can hear it's doing something, but too little to be noticeable in any real-word application.

So, what values do you recommend for a good all-around P-Bass?

Something I've noticed with the cheapo pots is that their audio taper is almost linear. No big deal for the volume pot, but turns the tone pot into an on/off switch.
Re tone cap, .05 is fine, though .1uf also works well on bass.

My wiring compendium is only in German: http://161589.homepagemodules.de/t29f2-Cadfaels-kleine-Schaltplan-Sammlung-fuer-passive-E-Baesse.html
But the are most historical wirings for Fender basses from 1951 until today (5MB PDF with more than 300 pages).

Audio 250k pots and .05 mfd cap are typical for a P-bass.
Even when the Jazz Bass cap and many other caps were changed to .022 mfd in the 1970s, the .05 mfd cap remained for the Precision.

So, the Duncan wiring shows the typical P-Bass.

... their audio taper is almost linear. No big deal for the volume pot, but turns the tone pot into an on/off switch.

Actually my volume pot is an on/off switch, and my tone pot is pretty much always on.

Actually my volume pot is an on/off switch, and my tone pot is pretty much always on.
Yup, many players use the volume pots only if there are two (or maybe more) of them to balance the PU's. But sometimes in a "hell of a gig" it's vital to turn the volume down just a bit and the way towards the amp is full of dangerous traps like cables, pedalboards and groupies laying around, so the vol knob on instrument nonetheless proves useful ;)