You're making it about you when it is really about the customer(both you and your coworker want the same thing). To most of us (hearing and non) what we do is just a job. What most people fail to realize is that the customers are really our boss. No kidding, if they don't buy the product you're waiting at the end of an unemployment line.
It would be one thing, if a boss/coworker suggested you wear a badge for no reason, but it is quite another if it helps a customer get what he/she wants. After all, the customer just wants his product, that's it.
I know this might feel difficult for you, but unless you think there was intentional harm between you and your coworker, I'd say the guy was just trying to help you out.
If you can find a better solution, I'd like to read about it. How would you handle the situation? There are going to be times you have to deal with customers, no matter how little.
In my case, when I couldn't hear well on the phone anymore, I would have a customer service rep talk with the guy and he would relay what was said to me.
In your case, that would work too, but I'm guessing you would just hand it over to the other guy anyway. The badge just further facilitates it. Is it less confusing? No, but neither will be trying to explain to the customer you are deaf, right?
Nobody wants to be labeled, but is it really any different than telling the customer you are deaf? After all, all roads lead to that conversation..