The tone cap will not do a whole lot to your mids. It will change the character of your Tone pot.
The pot values will adjust a little of the overall tone, but it will not squish that mid hump down. Seymour Duncan has an article about it in the website's Support section explaining how increasing the pot value affects the tone (it may be under a guitar type section, since I read it when making mods to guitars a few years back).
Changing the pickup may be the best bet, but as mentioned, the location of the pickup, as well as construction (single coil with it's narrower magnetic field, much like the Jazz Bass), as well as the bass' construction itself is what makes the '51 sound like it does.
A Quarter Pounder will give you a more agressive sound (more bass, a different treble character), an Antiquity will sound very similar but probably a little smoother on the top end, and the Rio Grande's will be similar to the QP's... I'd bet the QP's and Rio Grande's may even add more mids to the mix.
The problem is not caps, pots, or pickups... it's the character of the bass in general.
So unless you want to route the wood for another type of pickup entirely (Jazz, P-Bass, or other pickup), you'll probably just have to tweak your amp's EQ.
The look AND the scrappy midrange is why I'm a huge fan of the '51 RI's myself.