You are not being unreasonable to say cash payments are preferred, but I would think you would lose business if you tried to enforce cash only (which you haven't said that you will).
If encouraging cash payment you need to make sure you have a decent float to pay the correct change.
Our local chippy posted on Facebook explaining how much they had to pay in bank charges, and asked if people could pay by cash if possible. There is also a small sign by the till to say something along the lines of card accepted but cash preferred. ( They have since had a cash point put in- not just for customers, for anyone to use. There is a cashpoint not far away, but if I needed to use one, I would prefer the one inside the chippy, to the one in the wall outside the local supermarket).
I wouldn't take offence at it. If I had the cash available I would pay that way. I would prefer that to being told i could only use one payment method.
Different businesses will have different things that work for them. At a guess some prefer cash due to bank charges. Others may prefer card because they don't want to have to go to the bank with cash, no risk of fake notes, no mistakes in change, could be easier for accounting purposes.
I think I would change the wording a bit to something like-
As a small business we are charged by the bank for every card transaction. These charges add up each month.
If you are able to please consider paying by cash.
If you are not able to then card payment or bank transfer are also accepted.