The best way IMO to get good sound out of not so expensive bass equipment is using Tech 21 Sansamp as DI box.

Bass > Sansamp > Firepod > Record > EQ

i have an american P-bass but i've heard the mexican ones are pretty good.
im not a natural bass player so i've had to do a lot of experimenting and practicing concernig "muddy sounding bass".
i found 3 things anyway that really matter and cleared things up for me.

1-a good DI box matters even if your interface has hi-z ins.(followed up with proper e.q.)
2- flatwound strings are easier to play and record. they also sound tighter,less noisy,less muddy.imo
3- playing ability matters more than the previous two.
my 2 cents..

All correct.

Recording bass is more about the player than the instrument itself.

Flatwounds do record better for most styles.

You can replace the pickups easily and if this appeals to you you should do it. Seymour Duncan's or Wilkinsons.

And no amp.

Agreed with all the above. The MIM (made in Mexico) fender basses are good instruments. The biggest "problems" with them - the truss rod and pots seem to fail more frequently than those in the MIA basses - are not an issue regarding sound if those parts are in good shape. I like to swap out pickups. I'd add the Fender Vintage '60s p-bass pickup to Dave's list. But as said - tone is in the fingers. It's mostly the player. Now, in order to play well it's best to have a bass that's set up to your playing style. Neck relief and string height matter a lot. The nut needs to be correctly cut. The frets should be level and polished. The strings should be in good shape. (Another nice thing about flats is that they stay in good shape forever and put much less wear on your frets and fingers.)