Custom Resource Packs are only recorded when playing on a multiplayer server.
That's simply due to Minecraft not sending those Resource Pack packets when playing locally. Not entirely sure but I believe when you join another persons' LAN world, you don't get to see their resource pack either.
So if I were to host the replay on a server, download myself the resourcepack (as a server resourcepack), then play the song, it would work?
EDIT: Seems that the mod I was using does not work for multiplayer. Worth a try :)