If the crossover on the BB2 relates to the Super Twin in a similar way as the Super Midget to Super Compact, I'd hazard a few guesses based on my experiences with the BBBOD, Idiotbox Blower, Agro, and W&C TFR.
Turning the tweeter/horn down isn't exactly like not having a tweeter/horn, but it can get pretty close. At different volume levels, rooms/settings, and EQ settings and tone goals, it can be nice to dial in a bit of extra tooth with the tweeter/horn. I don't find myself turning the tweeter on the SM off, just down. I also tend to run it with my Demeter VTBP 800D vs. the brighter Mesa Buster pairing with the Super Compact. The cleaner the sound, the less bothersome the tweeter brightness is (in fact, it is nice, and feels integrated with the woofer sound). I've had SM/Demeter with the treble up in the bright setting with a J bass (Spector Forte SS), and liked the tone. Adding more distortion certainly makes the added treble a bit harsh and pokey, so the tweet, or the eq, or both will get adjusted down. The Blower is very toothy/bright and the pedal EQ almost always needs to be all the way off. Brighter settings (north of noon) on the TFR get way too crispy and sizzly. The Agro is somewhere in between; it can get more top heavy as you turn the presence up and toy with the contour and gain.
Bigger box on the BB2 (probably similar to the bigger volume of the SC vs.the SM) to accommodate to horn will make the BB2 low end a touch bigger, which may help keep some balance with the potential added brightness of the horn.