Let's translate this:

"A client is purchasing a new car under personal name for business use"

Is a "POV" = personally owned vehicle, then.

"and company will pay for it."

Nope.

"What is the best way to go about that?"

The company pays this person wages through payroll. The company can accept a business mileage report under the terms of An Accountable Plan and reimburse per the IRS mileage rate and this is not taxable to the shareholder-employee.

"Would section 179 and bonus depreciation apply in this situation?"

No. The Company doesn't own this vehicle. You stated that clearly.