"that building that houses the sheep" is actually the White Mountain Research Center,
Barcroft Station, run by University of California. There are researchers and students there all summer.
Hit it on an "open-gate" weekend, and it knocks off 4 miles and some elevation from the hike. In 2016, it was open
July 24.

AMS takes some time to show up, so there may be an advantage to getting up there, hiking to the summit and back down,
before symptoms have the time to grow.