The first thing you should do is to make sure the problem isn't caused by something else in your signal chain. Changing to a different P-bass pickup will (in most cases) only change your tone in a relatively subtle way. There are other factors that can influence your tone FAR more than the pickup:
-Your amp/cab (especially the cab). Have you tested your Tokai with different amps?
-You eq settings. Do you boost the lows anywhere in the signal chain? Cutting the lows (or low mids) a bit might help.
-The room. Do you experience this problem at different locations, or just in ONE room (like a rehearsal space)?
-Your choice of strings, or if the strings are old/worn.
-Pedals and pedal settings.
Having said that, pickups can definitely affect the tone as well.
There are LOTS of P-bass pickups to choose from of course. I never recommend gear I haven't used myself, so my list below is just a selection of popular options that I HAVE used, in my own basses, recorded with them and played them through my own rig, which gives me a good point of reference.
Here are my recommendations for P-bass pickups with tight lows:
-Lollar standard wind. Very vintage sounding, mid-focused, tight lows. Excellent pickup. One of my favorites. It really nails the classic, vintage P-bass tone. Very articulate, can get a bit "clacky" if you dig in hard (just like a P-bass SHOULD in my opinion).
-Fralin standard wind. Deeper, more extended lows (and highs) than the Lollars, but still very tight. A slightly smoother, more "modern" take on the P-bass tone than the Lollars, slightly less "clank" in the upper mids. A fantastic sounding pickup, also one of my favorites.
-Kloppmann PB63. Perfect in every way, perhaps my personal favorite P-bass pickup, but quite expensive and it isn't really any better than the Lollars/Fralins, just slightly different. Perhaps somewhere between the Lollars/Fralins tonally. Tight lows, perfectly balanced P-bass tone.
-Fender Originals. I think the are called Pure Vintage 62 these days. It's the ones with the beveled pole pieces, so they should be easy to identify. Very tight lows, but somewhat aggressive in the upper mids. On some basses, and with some strings, they can be a bit harsh/clanky for my taste, but I generally like them, and they're a very good choice for tight lows.
-Fender Custom Shop 62. I'm a bit hesitant to recommend these, because they are often critiziced for being slightly mid-scooped compared to other P-bass pickups, and rightly so. They're not bad, but they're not my favorites either. I find them slightly uninspiring compared to the other pickups I've mentioned, but the lows are definitely tight.
Another one that MIGHT be worth considering is The Creamery's 60s or 70s wind. To be honest I have only tested their 50s-something P-bass pickup. It's a fantastic sounding pickupc, but I think it might be slightly on the warm/fat side if tight lows is the main deciding factor. But their 60s and 70s versions should have progressively tighter/slimmer lows. The Creamery build all pickups to order, so you could even explain what you're looking for and get it made to the exact specs you want. They're surprisingly cheap as well, for custom made pickups. But like I said, I haven't actually tested the 60s/70s version, only the 50s version which is PROBABLY a bit on the "warm" side for your tonal needs.
Then there are two bordeline cases that may or may not be the right choice for you, even though they are great sounding pickups. These are the EMG GZR and the SeymourD SPB-1. Tonally these are both leaning slightly towards "fat" rather than "tight". Especially the SPB-1. You might like them, they're great sounding pickups, but for tight lows I would perhaps rather choose on from my list above.
The ones you should DEFINITELY avoid are any kind of "overwound" pickups, Fender Pure Vintage 63 (extremely bloated/muddy in the lows, personally I hate them) and the DiMarzio Model P (which I actually like, but it has very powerful low mids that can get a bit muddy sometimes, and also slightly lacking in clarity/highs compared to the ones in my list above).
You probably also want to avoid the Fender V-Mods, because they're simply not very good pickups IMO. The V2 isn't nearly as bad as the first version (which was absolutely HORRIBLE) but it has still strayed too far from the classic P-bass tone that I personally want from a P-bass.
Just my personal opinion of course. I'm sure others will disagree with at least SOME of this.