Travel to Mammoth, CA in Winter

The only passes open in winter are I-80 north of Lake Tahoe or Hwy 50 south of Tahoe.
50 might be dicey if there is lots of snow but I=80 is a major east/west freeway that is kept open all winter.....with frequent short term shutdowns.

Anywhere further south and all the passes shut down in winter due to snow.....Hwy 88 goes to the Kirkwood ski area so that might stay open for skiers traffic with chains often required.
For sure you will not be able to get over Monitor pass and to the east side of the Sierras.

88?? Not familiar with that route, in fact, never heard of it. However, 395 between Reno and Bishop is very unpredictable during the winter months. You would have to go day by day with the weather and pick one that is favorable. If you go over I-80 to Reno and then south on 395, the highway will be in the neighborhood of 9000 feet on several occasions.

The three passes south of Reno and over the Sierras during the winter are typically closed. That leaves you to go south to Bakersfield, over Hwy 58 (Tehachapi Pass) and then north on 395.....long way around but much less chance of hitting snow.