Appellant Can't File Special Response-to-Amicus Brief Under Federal Rules of Appellate Procedure
Any response must go in the normal reply brief.
So the First Circuit reaffirmed today in an order in Doe v. MIT, No. 22-1056:
Per the federal rules of appellate procedure, a party's response to the argument of an amicus is contained within its reply brief. To the extent appellant seeks relief, such as an extension of time to file his reply brief or leave to file an oversized brief, that request is denied without prejudice to renewal with the specific relief requested.
The analysis might be different in state courts that allow amicus briefs to be filed after the reply brief; but in federal court, this makes sense.
(Disclosure: The amicus brief involved was my own; I filed a brief in support of neither party, shortly after the appellant's counsel filed their brief, and the appellant's counsel wanted to respond to the part of my brief—to be fair, a large part—that is contrary to their position. I plan to blog about the amicus brief itself soon.)