I did some digging around in my 360's Video menu and didn't really find anything helpful for your issue. I was surprised that mine says it will output 1080p.

I had a large rear projection TV that I wanted to get rid of and a guy gave me the 360 in exchange for the TV. I never dug into the video settings since it worked, so I was surprised. The guy had another 360 and I wasn't looking forward to hauling a working 55" RPCRT display to the landfill, so it worked out good for both of us.

If you are sure your AVR setup is correct, the only other thing that I can think of is try turning the 360 on first, then the AVR. It might be that the AVR isn't detecting the video signal for some reason.