HDMI OUT 2 can be a tricky thing on any processor/receiver. Often, the HDMI OUT 2 port is not equally specced with the HDMI OUT 1 port, aside from ARC/eARC. Even when equal, a mismatch of monitor specs means output limitations on HDMI OUT 1 or no output at all from HDMI OUT 2.
I'd be willing to bet a difference in specs between the TV and projector is the reason for the issue here. I'd look at the incoming signal from the source device viewed on the TV using HDMI OUT1 and then also with the projector using HDMI OUT1. Do they match? A lesser specced display can ruin output for the other display depending on the ports used to connect each. HDMI OUT Zone2 is another can of worms.