First, I stand corrected re: TMLC being solicited by the Schindlers--their opinion was solicited by Jeb Bush. However, they do have a preferred outcome in this case. Their interpretation of Florida state law has to be considered in that light--if they make any error in assessing and interpreting the law, it will most likely be biased in favor of supporting a decision to intervene.
Second, they carry no more weight than any other group with an interest in the final outcome of the case. They have no official standing; they are simply another advocacy group. The attorney general's statutory (if not constitutional) duties include providing formal legal advice to the governor. If the governor acts against that advice, and is found wrong, he can be held civilly liable, and possibly criminally liable. If there are enough Republicans in the statehouse, then he'll avoid impeachment--but appointed officials involved in executing the decision who are aware of the content of the AG's opinion would be eligible for prosecution. And a leader has to think about the consequences of his decisions, including the consequences for those who follow him.