No, the V-1/Bottle Rocket does not have any transistors in it - it's a pure tube unit. (Even the only diodes are the rectifier, the LED and a couple in the switching relay control.)
The V-Twin is a tube/solid-state hybrid with IC gain stages in the signal path and solid state diode clipping on the Blues and Solo channel.
I agree about them being very different-sounding, but not at all about the V-1 being more smooth or scooped, or more 'transistory' - if anything, the other way round... which is not to say I don't like the V-Twin. I like them both, but for different purposes. The V-1 is a much more aggressive plain overdrive/distortion that works well into a normal guitar amp, the V-Twin is a more refined-sounding full preamp/amp simulator that works better into a PA, power amp or at most a very clean guitar amp.