Here is your issue or issues.,
On the road up to Big Bear are the CHP guys. If there is an substantial snow, they will declare a Chains ONLY day.
Then, if you don't have chains, they turn you around and send you back down the road.
So, if there is snow, you must have chains.
Now, the illegals are there to put your chains on for you , for a fee.
If you don' t have chains, some of them have those also, for a much higher fee.
If you have snows on, that won't help with the CHP.
It is chains, or nothing.
I would leave the box home and rent an Subaru. They do have a four wheel drive chain exemption under certain conditions.
Good luck and be careful out there.