"I have no idea which country you live in. But can you quote from any source were a competition has been advertised in such a way that makes the general public aware that a Ballroom Dancing Competition is pending."
Hi Three. I live in a fairly rural part of the USA. I know of a small private university in the area that holds a dance competition which has obtained advanced coverage in local media. So, to answer your question, yes, it happens.
But you may be missing the point. No amount of advertising or media coverage can "make" the general public aware of much of anything. All such media can do is put the information out there. People will read/watch/listen to things that catch their interest; they'll only retain things they read/watch/listen to if they value the information for future use. If the public doesn't CARE about ballroom competitions, a media blitz will be ultimately unproductive: your own comment about using Breast Cancer as a stalking horse/cause celebre is an indication at least part of you already knows this: if you can't get them to come for the dancing, get them to come for pink ribbons and a warm fuzzy feeling. Such an approach might work once or twice, but it doesn't address the underlying problem. MAYBE if people attend (instead of just buying tickets for the tax deduction), their interest might be caught -- maybe not. It's nice to think it would, but the public is not likely to have any greater appreciation of what they're seeing than they would a display of, say, fencing, or figure skating.
I would suggest if you want to get the public to come to an event, you first have to get them interested in dance, and then get them interested in competitions. And I mean in both the long- and short-term.
There IS an increased interest in dance, due in part to movies and TV (e.g., "Dancing with the Stars"). Personally, I've always thought that the latter has always missed a bet by not including a serious two-minute instructional segment in the show (I do remember a "comic" segment featuring a guy dressing in drag to learn Waltz, which, IMHO, flopped both as comedy and as instruction). Maybe DWTS has started doing more, it's been a while since I've bothered to watch it.
However I DO teach some beginning Community Education Dance programs, and while overall enrollment is small, a significant number of students enroll because of DWTS. So - the interest can be kindled.
If you want to try use a cause as a stalking horse to boost attendance, I'd say great, go ahead (I might suggest a cause related to a healthy heart rather than breast cancer, because the connection to the sport would be stronger). BUT - I'd also try something like throwing in a few free lessons or instructional demonstrations throughout the course of the evening. Anybody/everybody gets to watch the demonstrations, where somebody teaches how to a box and UAT in the waltz at 7:30, or the Tango basic and La Puerta at 8:00. They get raffle tickets with their admission tickets (free, or maybe for an extra buck), and every half hour, hour, or whatever, a free lesson is offered for 20 couples -- chosen right then by random draw of the raffle tickets. The lesson is given right then and there, in front of anybody else who wants to watch. The suspense of who gets called for the raffle tickets helps keep people there throughout the "show." You wouldn't want lessons/demonstrations to conflict with competition sections -- but maybe have the tango lesson just before tango competitions begin. As a long-term approach, maybe a local studio(s) can be persuaded to offer discounts on lessons to those who retain their admission tickets.... of course, some care should be used with this last idea, as some studios might abuse the situation (maybe a Community Education program instead?

).
Do something like that -- getting some people DOING (and having their peers watch), help everybody appreciate what they SEE the competitors doing, giving them something for FREE that whets their appetite for more -- IMHO, that's the only way you'll get any lasting result, any lasting increase in the general public interest in dance.
JMHO, for what it's worth.