Ah the best of the best...especially if you want to do a purpose-built custom.
The K1 is the first model to have the integrated engine cradle, welded inside the frame, meaning no possibility of stress cracks developing. It also has the rear brake cable mount, which went away during the K3 model run. And...it's not federally-required to have turn signals, a RH mirror, etc, that's all up to you, i.e. purely optional. K2 is the same but, they've rarely been seen; in my experience, they've accounted for 1%, over the last 2 decades. K1s are comparatively plentiful, second only to the staggeringly popular K0 model...which spanned 3 years(!).
IMHO, starting with a bare frame is a win-win. You get to build exactly what you want and that's one more CT70 back in action...which ought to hearten the purists, too.
Congrats, looking forward to your new project.