Contact the healthcare branch of the teachers union. They usually have a separate section just for nurses and it has nothing to do with a nurses degree. Also, try contacting the NURSES branch of the service workers union. They usually represent all the workers but have a separate professional section for just the nurses. Id go higher and question them re: what you have been told.
You have a right to be in a union and you have the right to choose which union that will be. The employer has no say in whether he wants "to consider speaking to a 3rd union" or not. If you become a union - he has no choice but to speak to you - & whichever union you choose to align with.
The ANA does not unionize you no matter what the size of your group. The role of the ANA is to be in Washington DC handling politics and policies that affect nurses.
If you want collective bargaining thru your professional RN-only organization, you go thru your state association to unionize. Since they represent RNs in any setting where nurses work, some state associations have some bargaining units that consist of very very few numbers. In fact, an entire bargaining unit consisting of just 6 nurses in the whole facility in Hawaii went on strike this year over staffing.
State associations represent school nurses in schools
all over the states.
Florida Nurses Association does have an RN labor union branch and you can call them to talk to you and your colleagues and give you information. It shouldnt matter the size of your bargaining unit. And you can call the offices of the NURSES branches of the unions that are already at your hospital. Call them all, meet with all, and compare what they have to offer.
Visit their websites, interview them, and ask lots of questions. Shop for where you will get the best representation to help you become the strongest union you can be.
Bear in mind that "a group" of workers has none of the legal rights, legal strengths, or legal protections that a "union" of workers have according to the law.
You can be a "group" and take your chances (which is not much different from what you already are doing) or you can be recognized legally as a union and have the power of the law behind you to have a say in your workplace environment and conditions of your employment.
Good luck in your decision.