I also don't know of anything that one might call a climber's route on Moosilauke. However, if "running legs" suggests that he might prefer to run, the Carriage Rd might be worth considering. Park at Breezy Point and follow the road up to the ridge between the South and Main summits.
The road is used as a ski and snowmobile route in winter. I haven't been on it in summer in quite a while--others might be able to give you more up-to-date condition reports.
Doug
Agreed. The lower part of the Carriage Road might be a bit wet, but the upper part should be fine, and overall probably the best route for running rather than hiking legs. I will be doing trailwork on the Glencliff Trail on the way up and down, so can look for Hugh at the intersection below South Peak.