Ok this is what I learned during a copyright infringement a long the way.
1. Copyright in the USA doesn't apply in other countries. If hey have no copyright in the country where the business is run, they are bullying you. The record label I worked for had to have a copyright for a song in each country.
2. A simple statement on your website saying you are not associated with their product may help.
3. When Wendy's started using "Super Salad" a term used by our restaurant for years and copyrighted by us, it was simple. Wendy's said you can take the small settlement and let us have "Super Salad" or we can tie you up in court. We have staff lawyers and we can tie up in court until you are broke. Wendy's got "Super Salad".
So if they do have a legal claim on your location and appear to be big enough to sue you, give up and change the name.