White Mountain peak is mystical. It looks like a martian landscape and seems like it should be a breeze to walk up, given its gentle grade... but WMP is the only peak I've ever had full blown mountain sickness (twice!).
Maybe there is something about being jostled around on that horrendous trail for 2 hours that makes acclimatization more difficult. Sleep at 12k ft is not usually an issue for me, but is always awful for me at Barcroft area. I don't know why. Headaches are massive and striking sleep is difficult and nausea comes and goes. But, suffer this and you'll get some privileged scenery. The views from the summit to the west are epic, and occasionally you'll see some gnarly old bristlecone. Also, you'll likely see bighorn sheep and a peculiar black stallion that apparently escaped from Bishop years ago and haunts the hills up there.
The roads are known by the locals as particularly problematic, and they are often called on to help for flats tires, etc. Don't expect help. Tow trucks can charge over $1000. I'm pretty convinced that the best way up and down that mountain is on a mountain bike--and I wish I still had the fitness and technique to do that. Another way is to hike it winter and do the 20+ miles on foot. Just be aware that wind across the summit can exceed 100mph, and you'll need to melt your own water from the airy snow that accumulates. I did this many years ago and it still is one of my favorite trips.