Why can't you not accept their card?
B) Do you want to be told - "sorry, I can't accept your BoA airline rewards miles VISA card, please use a different VISA card"? I'm pretty sure Visa/MC TOS prevent that.
Why would you do that? As far as a vendor is concerned, a visa is a visa is a visa. If you take a look at the card and say you don't want to accept it based on it being some kind of rewards card that's just bizarre nonsense that would definitely keep customers away.
Nope. A "rewards card" CC jacks up the rates I'm charged. It's the difference between paying 4.5% to process a transaction vs 2.2%. I don't particularly like being forced in to paying extra so someone can go on a 'free' airplane trip. It's not free. It costs me money. And it ends up costing everyone even more, because merchants have to add that 'cost of business' in to the price of everything.
Refusing AMEX is a big one from many vendors - AMEX fees typically start at 4-5%, and MC/Visa is typically 2-2.5%. On large repair bills of, say, $1500, and extra 2% is another $30+, just for accepting a different piece of plastic.