I've never played a Barefaced cab (though I know they have a good reputation here and elsewhere) and I never play with compression, but I have both a Monique and a Walkabout and I did a little reading on the
Super Compact and the
BCP.
While Monique does some things better than any other amp I've got, it's not the brightest sounding head, and can sound thick and dark when played back-to-back with some of the others (but then they can sound thin and harsh played back-to-back with Monique). The Walkabout does have a more flexible EQ and a more open high end than does Monique, and the Super Compact is a one-way cab (though I don't know how it's voiced). I can't say whether the combo of the two will work for you, but here's what I'd try:
1) Play with the Sensitivity control. Aside from affecting feel, it has a significant effect on the harmonic profile of your tone. Turning it up can add texture and detail to the sound, as can pulling the knob out.
2) Play with the EQ without preconceptions about how it should look and see what you find. Everyone says this, but it can be hard (for me, anyway) to do. On most amps, I boost lows and low mids and favor the bridge pickup when I play. Through some rigs, it works better if I favor the neck. With Monique, through some cabs, I might cut a bit of bass. I tried it for the first time through an Epifani PS-112 today, playing an occasionally dark-ish sounding fretless bass. I got my best tone with the Lows at noon, Mids at 1:30, Highs at 2:00, Sensitivity at 12:30 and pushed in. (My usual settings with Monique: EQ flat, Sensitivity at 10:30 and pulled out.)
3) I'm guessing that the "tilting EQ" on the BCP pedal was more responsible for the sonic help you experienced than the bit of compression you dialed in. If so, you might try some EQ in front of Monique -- or between Monique and your poweramp, if that works in your setup. I've considered getting an Empress Para-EQ, myself. I mentioned in an earlier post that I liked the effect I got running Monique through the effects return jack of a Demeter HPB-1 800D amp and dialing the presence (a gentle, shelving boost starting at 2 KHz) up.
Trying to think of what "congested" might imply. Maybe veiled up top... maybe too thick in the mids/low-mids. Try dialing the bass back, and see what you think. I guess an HPF like the fDeck is a possibility (it can "air out" some setups). You could try dialing in a tone you like with the Walkabout and see if you can come up with a version of it that you like with Monique... you won't get exactly the same thing, but it might serve as a starting point. (Are you having to use very radical EQ on the WB?)
I'm just brainstorming, here.
Two things I've noticed in my own playing of Monique: Even if the sound is sometimes not quite as bright as I'd like, chords and harmonics still speak well and clearly -- sometimes better than with brighter sounding amps. And I'm not afraid to solo the bridge pickup.
It's possible the cab and amp aren't the best match for your tonal goals, but that's what I'd try. Good luck!