I don't know what the multi-comp is like, but if you can set the attack time a little later, that might help. Also, blending is a good suggestion, but I don't think it would make it ok to have the bass drive on the whole time. You could have a few different pedals in the blended loop though.
I would suggest buying a good low gain OD pedal, boss ODB-3 or Marshal jackhammer or something similar, and put this in a blend loop at about 50%. Put a compressor with slow attack on your "clean" signal that you're blending with. Put a limiter after the the whole thing. You could leave the overdrive on the whole time, just adding a bit of "bite" rather than "dirt", and then you could use the bass drive in conjuntion with your new overdrive for when you want more dirt.
A Bi-amp setup would be useful, but obviously not essential - if you have a full range amp, you should be able to get the tone you're looking for.