For anyone else who comes across this problem, I managed to figure out a way to fix it.
The problem seems to stem from the OvrAvatarSDKManager object not surviving the new scene load. To fix this, once the first scene has been loaded, I find the OvrAvatarSDKManager object, and add a script to it that marks it as Don'tDestoryOnLoad.
To the Oculus Devs, would it be possible to include an option to automatically set the manager as DontDestroyOnload? Or does this already exist and I just missed it?