It's not everyone's favorite RHCP album, but I'll rock
By The Way all day over the others.
There are a couple options that would solve the blend issue. 99% of blends are passive, rather than active or buffered. Passive blends (for the most part, there are some miniscule exceptions) result in a volume drop when centered and/or some tone loss. Additionally, with a lot of passive blends the 'noticeable' effect of blending pickups together happens in a narrow range rather than equally over the full sweep.
One way to solve this is with a preamp that has an active blend. I believe EMG, Audere, and some others make this. There are a few threads about this. Here's one:
Buffering, or getting the "best" blended sound/level from passive pups
The volume-volume setup is really common with passive jazzes as you know. So you could go with a passive setup that has volume, volume, tone, and then depending on the pickups you get, a switch for single coil-parallel-series switching. This switch would give you lots of tonal flexibility and would definitely put you in jazz bass territory. This gets back to the question about what don't you like about the controls: approaching the same question from a different angle, are you wanting a passive setup where the tone is cut only or do you want active eq like bass boost/cut, mid boost/cut, and treble boost/cut?
Bartolini makes replacement pickups for the stock MK1s called MK6. Unlike the MK1s, which are just split single coil (like a p-bass), the MK6 is a true dual coil that allows for the single coil-parallel-series switching. They would be in your budget.
I know a lot of TB folks have installed Nordstrand pickups as replacements. I am not sure whether they had to do some routing though. Sets of Nordstrand pickups will be around $260. Check out bestbassgear.com for other pickup and preamp options.