"Would it be correct to depreciate it under the LLC (Schedule C), even though it is not under the LLC Name?"

There is no difference.

There is no "LLC" Sched C. The Sched C is a Form that is part of a 1040 Package. A 1040 tax filing is for the individual tax payer and your client happens to be running a business, by themselves, which is why you will use Sched C in addition to their 1040.

Forming a Limited Liability Company provides some business protection for an individual as far as State and Legal issues go. For Fed taxes, there is no separation: the person and the LLC are the Same Being. The Feds don't not have "LLC" entities for individuals.

What you might want to is direct your client to seek legal and insurance guidance as to whether the vehicle should be titled and insured differently. But you are doing Taxes, so for you, there is nothing different here. You have the same as Sole Proprietorship, because you have a Sole person running a business on their own.