Domain Advice / Name

Hey guys, I have a domain / business name that includes the word Phoenix in it. I named the business after the mythical bird.
Phoenix is also a city in the state of Arizona in the US.

I am not looking to rank my business in the city of Phoenix AZ, rather I want to rank it in the city of Houston TX. The closest I got was getting in on the first result of the 2nd page for an important keyword. I neglected the site and it completely disappeared from Google.

Not only does the business name have Phoenix in it, but the name itself sounds like a keyword. I know it's possible for me to rank the website for my city (Houston) but I'm starting to think it'll be a hassle / harder since it has Phoenix.

What would you do? Rename the business and get another domain name or keep on with the SEO?

For the record, I have put some effort into branding this company but I would not mind renaming if it'll be easier to rank, and have a name that people don't associate with another city.

The site does show for the name and in the maps since I've done citations.

 

Thanks for your input, it's reassuring me into what I already know

 

This will depend on your business and target location. If general terms brand+keyword.com will do or for specific targeting brand+keyword.uk for United Kingdom location target.