Whilst businesses are charged something in the region of 1% to 4% for credit/debit card transactions, they are also charged something like 50p for each cheque banked and even more if it's physical folding stuff counted and paid in.
So there's no saving for the business in dealing in cash and more inconvenience. The only way you could do it practically would be by BACS transfer via internet banking but this involves a time delay of 3 to 5 working days (unless you want to pay up front and you don't want to do that, do you?) and hence you'd need a credit account.
If I was the accountant of an FTO, I'd ask for credit/debit card and nowt else. Credit card is guaranteed; BACS not necessarily unless you did it there and then in front of them!
Cheers
Whirls