I'd be interested in:
Wet/dry paths
Separate stereo paths with effects assignable to each.
Pitch sensitive waveform generator with square, triangle and sine wave forms
16 step arpeggiator with individual ADSR settings per step
onboard signal mixer (6 channel)
Traditionally, most bass players arrive at a sound that works for them and stick with it.
Because otherwise the sound engineer will likely get on their case for making them faff about too much with the mixing desk settings! 
Shifting between pickups should be enough for most songs and use a comp after any bass synth pedals to limit the low end boost.
Jazz basses are great, but P basses have their uses (think Iron Maiden).
P basses tend to sit in the mix better because they have more mids, especially when strung with flats.
call me old fashioned. but find a bassman amp.. all is good
Pairing the right amp with the bass is important too. Often bass amps are intentionally coloured and it helps to get one that doesn't suck out frequencies from a place where the instrument is strong. I discovered Smith basses and SWR amps both have scooped mids, so scooped that I couldn't actually hear the pitch when playing live. I ended up keeping the bass and getting a GK RB700 instead as it was a bit more aggressive, especially in the upper mids. Both are gone now but that combo worked really well live. P basses and SVTs are a classic match, anything with a graphite neck into a valve head or SWR is great. Fretlesses need a LOT of mids to sound great (and to hear oneself). Jazz basses are pretty much the industry standard but they are particularly complimented by Mark Bass for finger style if you like that bridge burp. Or SWR if you prefer a Marcus Miller scooped snap. MM Stingrays go really well with valve heads - particularly the Mesa Bass 400+ and Bassman. But also with Warwick heads maybe because classic Warwick BASSES (like the Thumb) have a similar timbre to Stingrays.
It's for my live sound, not recording. Probably should have mentioned that in the beginning
Mesa Boogie Bass 400+ is my fave valve head of all time. It's not my backs favourite through. It'll do what you ask though and there are a few profiles in the RE. I would also suggest the Trace Elliot V8, I had one for a while and it had too much low end through a couple of EBS 2x12's. It might have worked better with a couple of 4x10's instead but hindsight is a wonderful thing.
Be careful in what you expect from pickups. They make more of a difference in guitar than bass, which isn't to say that they don't make any difference. Because basses are under greater tension, the materials and construction matter more, particularly when it comes to string feel and timbre. Classic Barts tend to be a little on the woolly side which is great for bright and rigid instruments, they will tame the highs a bit. They also tend to be a market leading shape but it's always good to to check as even EMG's and Barts can differ in width by a mm or two and that can make a difference to whether they'll fit or not.
I'm not familiar with the MK-1's in your bass but most of the Ibby basses I've tried have a prominent lower mid range and a slightly woolly upper end so replacement pickups might give you more to work with. But they won't make a good bass into a great bass.
Active EMG's are good for recording but they coarsen the attack a little if you're into hifi MM type crispness. You could maybe investigate their passive range though.
Many of the other soapbars (e.g. Nordstrand, Aguilar, Delano) tend to be pretty similar in sound and all of them are crisp and well balanced. You could also look at Seymour Duncan and Dimarzio but they tend to be for rock players if that's your thing. Q-tuners are super hifi and you can tweak them to manage where the resonant peak goes. Most manufacturers of flatter response pickups wind it below the threshold of human hearing.
Something else to note is that multicoil pickups are coming on to the market from multiple start ups like Turner. But they're not cheap due to the extra work in putting them together.
I use either an FEA Labs DBCL dual band compressor limiter for heavy squash on the higher frequencies or an Effectrode PC2A for more subtle taming. I haven't yet tried my kemper with my basses (only just got it) but am expecting dynacomp character.
Necrothread resurrect:
Try the Panda MIDI Future Impact bass synth pedal. Or if you really want to twiddle with settings, there's the Source Audio C4.
