You're a lawyer, so I assume you would know better than me. Would that not create a free speech issue? The arena and AT&T stadium are likely another animal since it is likely private entities having an agreement with a performer or a broadcaster, but I would think a government body telling people how to brand something on a public street would be an issue.
Like the Senate can and often does, it passes a non-binding resolution known as "A Sense of The Senate". Fort Worth could do something similar.
Wikipedia - a non-binding resolution is a written motion adopted by a deliberative body that cannot progress into a law. The substance of the resolution can be anything that can normally be proposed as a motion.
This type of resolution is often used to express the body's approval or disapproval of something that they cannot otherwise vote on, due to the matter being handled by another jurisdiction, or being protected by a constitution. An example would be a resolution of support for a nation's troops in battle, which carries no legal weight, but is adopted for moral support.
As to the "freedom of speech" argument, I do not think that it would be a constitutional issue. It is an issue of copyright for the City and I would say that the media behavior in suing it is more akin to the acts of a squatter - taking ownership of something that does not belong to you. This would also seem to be applicable when the NFL, NBA, NHL ,etc. tags Fort Worth as the territorial property of Dallas. In a one exception, the leagues do demonstrate a sense area inclusiveness by using Minnesota as the name of their teams. And of course, The Rangers are the best giving recognition to all in the North Texas area.
I do believe that organizations that tag the name "Fort Worth" for statistical and ratings numbers for monetary gain and then to conveniently omit Fort Worth at their convenience may be viewed as causing tangible and material harm to the City when their actions indirectly favor one city above the other.
If you tag "Fort Worth", then always included in your brand; if choose not too, at least ask those who used our name to be factual that the statistics and ratings that you provide are incorrect and misleading.