On reading that Florida State vows to keep its mascot, I thought of the Supreme Court's decision in NCAA v. Tarkanian, 488 U.S. 179 (1988), in which the split Court held that the NCAA was not a "state actor" for purposes of 42 U.S.C. § 1983.
So, FSU even it is an "expressive association" within the meaning of FAIR v. Rumsfeld (the Solomon Amendment case, now before the Supreme Court) would not have a constitutional remedy against the NCAA.
By contrast, I suspect that following Brentwood Academy v. TSSAA, 521 U.S. 288 (2001), a state high school athletic association could be sued under 42 U.S.C. § 1983, if it took on the naming of mascots.