Both the Main Trail and the Mountaineers Route start at the same trail head. Both require the same Day Use permit if you are only going in for the day. However, overnight use requires two separate permits -- Whitney Main Trail, or North Fork Lone Pine Creek.
As for skiing, most people will do that in February or March. Neither chute is very good for skiing, and this late, the sun cups would be a huge problem. The MR chute is pretty narrow.
The "chute" to the west of the switchbacks of the Main Trail is more a wide slope than a chute. It melts out completely, so rocks will be your problem there.
Davey at the Whitney Portal Hostel would be the guy to talk to about the skiing. They try it every so often.
