A dog that appears to be a service dog reacting poorly to strangers reflects badly on service dogs even when they are in dog-friendly places. SDs are allowed and common in pet-friendly areas, too. I'm only suggesting that you make it obvious the dog ISN'T a service dog. That's why I'm suggesting patches that say why you can't pet the dog, and a device that will actually help your dog if it's anxious about people.
And a bandana wouldn't really work, cause then people have to be right there to see the bandana
People have to be right there to pet the dog, so I don't see the issue. The patch is the same size regardless of what you put it on.
Gavroche is not good with strangers, either. I tell people not to pet him. I step in front of people if I need to or send him to stand behind me. It's our job, as our dogs' handlers, to make sure their needs are met. If that means your dog doesn't like being pet by random people, and you see a random person approaching, you make an effort to block that person from petting your dog. A patch might help discourage people from trying, but it's ultimately the handler's job to stop it.
Now, the vest Bahamutt posted is largely advertised as a convenience vest, not a service dog vest. I have one for each of my dogs because they're nice for walks (they're called Poo Boss vests, because they're designed to hold empty poop bags on one side and full poop bags on the other). They're also reflective and I do a lot of night walking. Speaking of which, how about a safety vest with a "shy dog" patch? They have some vests like that available at Petco. Bright orange or yellow with reflective stripes.
I also don't have a problem with a pet in a pet-friendly area with a do-not-pet vest if the dog isn't acting aggressive, scared, or reactive.
I also find "working dog" to be a misleading patch for a pet. "In traning" is fine by me, but "working dog" generally means the dog has a job, whether it's an SD or a SAR dog or what have you, and is actively working.