There is a law requiring Homeland Security to turn over their roll of non-citizens to the State.
The Constitution leaves the power of voter eligibility to the individual States (hence Montana being one of the first to allow women to vote).
Florida should just tell the AG to pound sand, and sue for the list. Continue to do what they are doing, and dare the AG to take action. If sued (in progress) have the court throw out the case for lack of standing.