I use both the Fulltone Bassdrive (2006 pre-MOSFET) and Fulltone Fat-Boost.
They both change or heavily color your sound depending on how you perceive it. They are both really thick in the mids, which I think comes from the “drive” part of the pedal. Although the bassdrive is capable of dishing out some serious distortion when used right, it is an overdrive pedal. I don’t know if this is a justification for the heavy coloring of the sound, but I have tried it with 5 different basses, and always with the same result. Low end loss compared to the clean signal, and a boost in the mids. The Comp mode is the least coloring of the 3 modes. But in general I think of the pedal as a mix of EQ-Comp/Limiter- Overdrive/Distortion pedal.
EQ - Comp/Limiter part.
It cuts all or a lot of spikes from your signal. This can be a real pain to control when balancing out levels. You can have the pedal turned on and the amp showing lots of headroom with no spikes. Going to clean sound with the same volume, your amp starts clipping because of the shift from heavy mids to heavy lows, and all of the normal “spikes” from your bass reappearing.
And to make life even more difficult, the distorted sound just sounds louder than the clean sound on the same level. This isn’t the pedals fault though, but just how frequencies and loudness are perceived.
Overdrive/Distortion part.
Comp mode: This is the most natural boost of the 3 modes I think. And it quickly becomes insanely loud. With the level knob at max I found that it could be used as a two channel preamp connected directly to my poweramp, with distortion levels controlled by the two drive knobs.
FM mode: modern distortion with heavy mids and loss of low end.
Vintage mode: Vintage

distortion with heavy mids and loss of low end.
All things said. I absolutely love this distortion pedal, and the fat-boost to for that matter. The sound is just amazing. But I have always had level problems no matter how much I turned knobs or EQ’ed. So for me it is an “always-on” pedal and as soon I get the money for a Blowtorch, it will probably get retired to recording pedal.
Again, this is such a great sounding pedal, and I absolutely love it. But live it just has to big an influence on your sound to be flexible the way I think most people would prefer.
Ohh… and don’t get me started on the “how to match the drive channel with the boost channel” issue….
I hope that you got something from all of this. And that you enjoy your bassdrive. It has a hard learning curve, but tons of potential.