I was curious and it turns out that the FEC does have regulation regarding public debates.
The part that would be relevant doesn’t seem to apply, though:
c. Criteria for candidate selection.
For all debates, staging organization(s) must use pre-established objective criteria to determine which candidates may participate in a debate. For general election debates, staging organizations(s) shall not use nomination by a particular political party as the sole objective criterion to determine whether to include a candidate in a debate. For debates held prior to a primary election, caucus or convention, staging organizations may restrict candidate participation to candidates seeking the nomination of one party, and need not stage a debate for candidates seeking the nomination of any other political party or independent candidates.
So, I guess there is a law that could be potentionally sued over, but I don’t see how the RNC doesn’t fall within the guidelines. I’m not a lawyer.