I'm guessing you also don't care for your distorted tones through the unattenuated tweeter in a full-range PA style bass cab. Essentially, what you want is to roll back the top end to mimic the limited top end of most bass cabs. (Some control of the gank range b/w 900-3K Hz would also be handy, but YMMV.) With a speaker simulation, the top end attenuation is baked in; with eq, you (or the FOH jockey) will be able to adjust to taste. FOH engineers can do this w/ eq, although *whether* they do so on any gig (and whether they do so to your taste) can be dicey. If it's really important to you, you could add a parametric eq to your stage rig and dial in a consistent DI tone to send to FOH every night.
Another option is to use a pre-amp/DI pedal for grit + DI. Some of these will give you eq or DI simulation; some will give both. This doesn't give you as much fine-grained control as a good parametric eq, but if you find a preamp/DI that gives you what you like and you're a set-and-forget player, then its ability to give you other tones doesn't matter much. For Wilkeson's SVT-era tones, you might have a listen to the Sans Amp VTDI or VT Bass Deluxe. If you're going more for the Peavey or Crown era live tones, you've got a bigger pool of contenders. (Especially if you're in love w/ your current grit pedal(s), you might consider feeding your dirt stomp box into a relatively clean preamp/DI. Likely to have to audition preamps through your rig, since the pieces will interact sonicly.)
