Is the bass guitar sound on "Whole Lotta Love" deep and full enough for yez? That was John Paul Jones on a Fender Jazz Bass with round wound strings and, I suspect, a plectrum.
Last weekend, I blew a large sum of money on a 2012 Fender American Vintage Re-Issue series '62 style Jazz Bass. I cannot tell you how good this instrument is without swearing.
*
LEFTFIELD SUGGESTION
If you like the idea of the Squier CV '51 style Precision Bass, find a pre-owned example and change the pickup. An active Stratocaster or Telecaster neck position guitar pickup would fit and sound much better than the stock item. (I have EMG-FT pickups in my Chapman Stick. The low end response IS there!)