He still has a choice, it just may not be a very likable one, or one you would ever want to have to consider.
For myself, personally, I would never be the one to take the risk of having to pay the business expenses of the company I work for. Since I would never have an AmEx card in my name (especially not for a company I didn't own) I would be left with the choice of looking for different work.
No, he really has no choice. The company pays the bills directly, and all we see is an itemized statement. Amex is the only company that the company holds a contract with to do this, ergo.....