We need a picture of your bike in that trunk!
You're going to get quite a workout on the MTB. The dirt road starts at Schulman Grove, about 10,000 elevation, and goes 16 miles, with 3400' of gain/loss, all between 10,000' and 11,900'
Here's the Google map.
White Mtn Road...that reminds me, I've always wanted to ride my mtn bike down the Silver Canyon 4wd road to Bishop. It tees off the White Mountain Road about 3 miles north of Schulman Grove -- 6000' descent over 11 miles.
Another option for you: Use a different vehicle, haul the bike to the gate, and then ride/push it to the summit. I once had a mtn bike on the White Mtn hut's roof. I was going to ride that Silver Canyon road on that trip, but the White Mtn ride completely wore me out.
