If it's anything more than free I'm sure not going to use it for something like paying a bill or getting coffee. My banks debit card doesn't charge me to do that.
If it's a negligible amount like 1 cent then people will think of it as free. Once it rises above a few cents people will choose to use free methods of payment like debit cars. The only case where people might be prepared to pay high fees is sending money abroad when the fees are less than the fees for wiring money between bank accounts.
For transactions where you normally pay a fee to send the money then Bitcoin should be a little below that but where there is no cost to send the transaction there should be no cost for Bitcoin either. If there's a cost no one will use it. You would have to be a fool to stop what you're currently using for free to move to something that will cost you to use.
Currently, the shop often covers the payment fee.. The fee, atleast in the Netherlands can be up to a euro per transaction (depending on your volume ofcourse). As a costumer, you don't see this because the shop takes care of this. Maybe an option like this, where the receiver pays the fee, could be an option?