before you worry about getting a fat libflashplayer.so there first needs to be a 64 bit libflashplayer.so for you to use and merge with the 32 bit one.
users of 64 bit desktops still user the 32 bit libflashplayer.so run though a ndiswrapper layer.
so this is not a case where FatELF would help in practice.
even in theory, firefox doesn't have to have the plugin binaries under ~/ so if you don't install them there and instead install them in one of the other places that it can live you would be able to NFS mount $HOME without a problem.
the same thing goes for any application that uses plugins. the plugin binaries should be able to be installed outside of $HOME. If they can be, then the application can work if $HOME is shared.