The only species of bears still existing in California is the common "black bear." The last know "California grizzly bear" was killed just after the turn of the last century in southern California.
One interesting tidbit about the California grizzly is that it was bigger than the Alaskan Brown Bears. It had the salmon for protein and growth, and the mild winters to stay healthy.
One of the things the Fish and Game learned many years ago with problem bears was that relocating them doesn't solve anything. Bears traped in Yosemite and released in the Yollo Bolly's would find there way back in a year or two, or get killed trying.
For awhile, they would say they were relocating the animals, but didn't actually do it. It is cheaper, easier and more effective to just kill the problem animals once they reach that classification.
Your bear under the cabin sounds like it falls into that category. If you get it to leave, it will come back for the rest of it's life. There really isn't any other alternative to dealing with an animal that has learned food can be found inside buildings.
The Fish and Game doesn't give out depredation permits lightly. They have every wacky environmentalist group breathing down their necks to justify each and every one they do give out.
They are also not equiped to kill the animals themselves, unless it becomes a self defense issue. Most problem animals are contracted out for elimination through trappers and licensed houndsman.
It might be easier if you look at it more along the lines of a very large varmit that has made a home under the house.
Good luck,