The point of putting assets in an LLC is often to protect your personal assets from your business liability. If you add another company to do business under the original LLC, you are potentially exposing one business to another business' liability. It would probably be better to separate the LLCs. That being said, in order to do business under another name in New York for an existing LLC, you need to file for a Certificate of Assumed Name. This form is $50 plus $25 for each county you do business in (except the city of New York where it is $100 for each county).
The best way of protecting the idea is probably to ask the networks to sign a non-disclosure agreement. The name is probably not the important part of the concept, although you can trademark it. It would be a good idea to discuss this with a business lawyer.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on since each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.