michael,
to answer your question you can write on the back of the check
"pay to: _______ (fill in the blank) and then sign your name.
otherwise you certainly could deposit the check in your own bank
account and settle with your shop using whatever method is acceptable to them - in your case if you earn FF miles from Amex
then you could earn a few there..
just keep in mind the merchant is charged a percentage of every sale by the credit card companies. Merchants' accept credit cards for the convenience of their customers and should be counting on paying that percentage when setting their prices/fees.
whatever you do check with your shop
before you endorse the check over to them!