"but now they are trying to argue they can have their business make the truck payment and have it deducted as marketing/advertising."

I got this question in every business financial management class. I pointed out that if they Park the vehicle, they can take Actual Operating Expenses (which is pretty much $0) as a billboard. If they intend to use it as a vehicle, it falls under vehicle rules.

And of course, they can deduct the signage, under the rules for something of that cost and useful life.

"Is there some other tax reference I am not finding that would entitle them to have their S-corp pay their personal truck payment and classify it as "marketing"?"

They can have their S Corp pay them anything they want to, subject to if there is another shareholder-employee who has difficulty accepting the reality of your client paying themselves outrageous payroll, of course. They don't need "tax rules" to follow, to set their pay at "whatever I want it to be, as long as I don't bankrupt the company." And that caveat never stopped some people, anyway.