AFAIK, the Fed side would tend to be backed by assets of entities 'whose assets are controlled'. The Treasury side would be backed by the gold. Perhaps you have misunderstood the distinction between internal workings of the U.S. Treasury (public money) and that of the Federal Reserve System composed of twelve Regional Reserve banks, the FR Board of Governors, etc. The idea of redemption of FRNs for lawful money is to go redeem the FRS clearinghouse credits/certificates for lawful money (i.e. Treasury side).
You might see if you can find a copy (online) of: Clearing House Loan Certificates and Substitutes by James Graham Cannon (funny, St. Louis Fed is now hosting a copy since I first mentioned it on this site).
Related:Attachment 2685
Does that in the slightest way look familiar?