As one of those LA Movie types, I'll weigh in.
There are several Picture Vehicle places that rent here in LA. The $$$ adds up fast. Typically a car like a W123 would require a "Certificate of Insurance" listing the owner as "additional insured" and "loss payee". The deductiible on the policy would be handed to the owner as a "deposit check" which would not be cashed unless there was damage.
It is common to demand that the car be driveable by an actor or actress. For which I have a funny story. I was doing a shoot for Playboy and the playmate couldn't drive a stick, so the '56 Vette was pulled by a rope into the shot. Driving off into the sunset was the Producer in a wig.
That said, older vehicles would typically not be expected to drive up mountains and then across the desert. That would be asking for a breakdown to hold up shooting. The folks with REAL money have 2 or 3 of the same picture vehicle. If you pay attention they don't always have identical cars and this will show up as mirrors that disappear and re-appear, etc. But I once fought long and hard to get an old VW Bus out to the desert to the 4 Aces location. Having it flat-bedded out was going to cost a fortune and ruin my budget. I finally convinced the owner to drive it there, and it ran fine. Cantankerous spokesman for Danish potato chip company drove it with no trouble in and out of shots.
Last time I went to Big Bear it was switchbacks, rain, and snow and we got stopped by an officer due to a nasty wreck ahead. We just had a bunch of snow last week in the mountains, so I imagine that it is still there. They may have decided to shoot now to see this, in which case a white car won't be ideal.
Anyway, I rambled. A picture vehicle company would charge $250-500/day for such a vehicle. They would also charge for a driver/handler and transport of the vehicle itself.
Here is an example:
Cars | Picture Car Warehouse
Whatever you do, I would get that deposit check up front, and something in writing about damage. (both body and mechanical) Just about anyone running a shoot in LA absolutely HAS to have liability insurance and getting those Certs was usually just a fax or phone call to an office somewhere.
And if there is an issue where you have to hold something to the left while pushing type thing, make sure you tell them, and maybe put labels on things that aren't obvious.