I would first suggest calling RC and speaking with a tech regarding the specs on the 6" lift; in particular, the new front knuckles. I'm fairly confident that most of the crossmember full lift kits increase the track width 1-2" mainly due to new knuckles. A clue to this could also be the tie rod ends that come with the kit. Are those longer than stock? If so, then track width has increased.
The 4" & 6" BDS Suspension lifts increase the track width, allowing for wider tires on stock wheels. I'm pretty certain the RC 6" lift is the same. So, YES, you will be able to run the 12.5s on your 20s.
I think most lifts are spec'd with 20" wheels, very common on lifted trucks. The REAR track width does NOT increase. If you want the front and rear to line up when viewed head-on, you'll probably at least need some wheel spacers in the back.
One thing to keep in mind, if the track width does increase and you run the wide tires on OE wheels, then you will certainly have outer edge of tires extending beyond the fender flare. That's illegal here in Maine, and could cause problems come inspection time. Not to mention all the debris getting slung up the side of the car, and junk getting thrown at others behind you on the road...unfortunately not too many options out there for our model years in the fender flare dept.
Also, with the 6" lift I would suggest 34-35" tires; a 2" level will allow 33s to fit. The lift and larger tires will certainly be a drag on your power. Not sure what gearing you have. I would suggest considering overall tire weight when making your final decision on which tires you ultimately purchase and install. Perhaps a 34x11.5 would be a good wheel to consider vs the 12.5?