I don't understand the need to route the video signal through the amplifier. The only benefit is that you don't need to change inputs on the TV when switching between video sources but that is just one button press so no big deal.
Two alternative options
1) If using standard audio amplification with no digital processing then route video from the player/tv box straight to the monitor via HDMI and make all your audio settings in the player and connect to the amplification via analogue outs
2) If using an a/v receiver with digital processing route video from player/tv box straight to the monitor via HDMI, route audio straight to the a/v receiver via digital coax and make audio settings in the a/v receiver
Current Lash Up:
TEAC VRDS 701T > Sony TAE1000ESD > Krell KSA50S > JM Labs Focal Electra 926.