I found over the weekend that Ulti X-Soft and X-Super Soft compound tires are pretty decent on black carpet. Fast and forgiving without getting tippy, with consistent grip and steering during the run and acceptable wear rates.
It's funny trying to run Mod on black carpet, as you soften the front end of the car it gets better and better until the chassis starts dragging. Initially I found that the car scrubbed too much energy on turn-in, losing speed and then being slow finishing the last 30 degrees of the corner because you had to give it some throttle to keep the car moving, but a swap to soft tires and front springs helped. Not perfect, but it was better.