2 hours to go riding would be great! From L.A. its more like 3 for non-plated bikes. As for Big Bear don't know, there are trails outside the ski area that I'm sure you could ride on. We ride dual sport up there quite a bit. Holcomb Valley has quite a few green sticker trails/roads that should be snowmobile friendly.
Now here's a crazy idea, snow bike with a plate!