Sign out of home sharing on all of the devices (Mac, ATV, iOS). Then, sign in again. Turn off the Airplay authentication on ATV until you are able to get a good connection, then turn it back on again if required.
I did the dual authentication a few weeks ago.
I've gotten it to work since once then.
Went back to use it last week and it didn't work.
Can't think of anything else I've done.