Go red. If you do blue, you will be bushwhacking a lot. Red has now been cleared and marked for large portion. Green bit is steep drop / side-hill to saddle below Nutt south. It is bit awkward but dog should have less issue with it than human.
I have not been to Nutt north, but John has dragged Katie (or was it the other way around) & they had no problems.
So, in summary, I should go red then attempt green and then follow blue to summits. Then retrace the route coming back.
Katie and I have done this route quite a bit and she has never had any problems getting herself up. No trail or markers after red. Bushy in spots, and one short scramble section just as you start the ascent to the true peak.