Sua sponte referral of postjudgment motion to another court is within the inherent authority of the court.
Whatever the legal status of the fee agreement, error to deny fees under S1988 where petitioner's primary interest is the vindication of the claim and the award is not de minimis; the fact that the award was under the last settlement offer does not negate the basis for the award.
http://media.ca7.uscourts.gov/cgi-bin/rssExec.pl?Submit=Display&Path=Y2018/D06-29/C:17-1876:J:Kanne:aut:T:fnOp:N:2179169:S:0
Since 2011. An occasional commonplacing of the U.S. Circuit Courts of Appeal.
Neither an authoritative statement of the law nor legal advice. Quick work.