Someone else called it - that the BR instructor is competing instead of cooperating.
Let's separate fantasy from reality: In the fantasy world, businesses always cooperate to find win-win solutions. In the real world, your competitor puts posters next to your studio.
Frankly, this looks likea zero sum game to me - IME there will be little overlap of dancers between AT and BR Tango, and the business will have to go this way or that - there isn't much of a middle ground. Your competitor seems to get this.
My advice - stop looking at community-oriented solutions as your primary response. Do what you need to do to create a civil and friendly atmosphere in the scene (worst case - you'll come off as the nice person), but for things like county calling you about posters, tell them the truth and redirect them to the right person (but do give the other guy fair warning that you're sending the county folks that way). Then figure out how to out-compete. Start leveraging social media and marketing tools first for e.g., - set up Meetup.com events and take ownership of all plausible Tango related names/phrases, get yourself on Facebook, use Google adwords and ensure that you are on top of the search results for Tango or dancing in general (pay a professional if need be to get these things right), play into Groupon if you feel like it (but evaluate it's flip sides carefully - groupon has a nasty reputation for bringing one-time clients, and few repeat-clients)... there is a lot you can do to up the game, and upping the game is the only way you can really create lasting peace. Establish yoruself as the top-dog, then extend an offer of peace - on your terms. Right now, you seem to be on the defensive.