my appletv is conected by hdmi to my tv
by optical to my receivers optical input
I mute my tv to about audio from 2 sources
I set my receivers active output source as the optical port in my case it call it BD because it's sony and they like blueray...
and it plays
some people repport that the hdmi have to have an active connection or the optical will be muted
when you troubleshoot you start with a simple setup rather then a complex one so ditch the projector maybe it's the cause connect a tv and the "another route" and test if it dont work then exchange the "another route" with a simple receiver and test again sooner or later you learn if your equipment is defect, incompatible or if your set it up worng.