Ok even using a sub standard Peavey bass amp it is possible to get a decent tone.
Set up a DI and def a mic. Something like a D112 or Beta 52A or if you dont have something like that use an Sm57 or Sm58 or something similar.
Place the amp up off the ground on a table or something. Place the mic about half a foot away from the speaker and find sweet spot by moving about the speaker.
Compress slightly on the way in if your bass player aint got totally solid dynamics. and compress alot more at mixing. Boost at around 3-5k and 100-160Hz (rough ideas).
Dont use a POD please. A DI'd and (mic'd) Peavey bass amp will sound well better and more realistic as the DI will be going from real amp circuitry and the mic (because placed back a bit from the speaker) will pick up some depth and beleivability.
Good bass guitars to record are in my view usually quite bright sounding guitars, like
the Fender Jazz bass.
Active basses sound better usually because the pickups are powered. Make sure if you have a panel on the back of your bass guitar that you change the battery regularly and especially when going to record, becuase a half dead battery will make the bass sound muddy and lifeles. Along with new strings your bass tone should sound real nice even through a Peavey bass amp!
Amp settings i would go for is take soe lo end off so you dont get any deep rumbles from the speakers, boost the highs and play with the mids till you get the clearest tone that still has body. (ie not thin.)
Make sure you dont forget your bass guitar tone settings. I ususally put all tone knobs up full. Try that and if the tone is a bit too high just lower the tone knobs a little.
As you were saying about 90% of the tone prob comes from technique. Using a nice hard thin pick where posible will give you more definition and calrity and punch. Although not always possible ie. (slap bass)
Good luck.
Its a hell of a lot easier than recording electric distorted guitar anyway.