I would assume a business card would suffice, no?
As for the insurance issue, if on personal business, the employer's carrier would deny the claim, and employee's auto policy would pick it up (assuming it was a short term rental and not a car that should be listed on their personal auto policy).
That doesn't mean one might be violating their companies travel policy, but it shouldn't get the company crossways with their insurer. The bigger issue comes from declining the insurance your company provides, as it may have much higher limits than yours, your personal policy may exclude business driving, etc.
If the employee doesn't have their own insurance then they better purchase it through the rental car company on personal travel.