Veterans Club with no more than 6 to 8 patrons on a given night
That would depend on your local health codes and the bar owner's individual policy. While a bar is a place of public accomodation, it is still a private business. If you have a service animal, that is an entirely different issue, but if it is merely a housepet, the bar owner's rules apply.
Most likely, the bar is privately owned and the owner has the right to choose the acceptance of pets on the premises.
You call it a public place but in actuality it is a private business and can decide whether to allow dogs or not putting aside whether Health Code regulations prohibit animals. However, if it's a service dog accompanied by someone who is disabled the dog probably has to be allowed under the Americans With Disabilities Act.