Do you feed anything else to the TV through the AVR via HDMI successfully? It certainly sounds like a handshake / HDCP issue between the TV and AVR or AVR and BD player but the fact that it's had this problem with two different BD players and two AVRs makes me question the TV.
I'm not too familiar with Sony TVs and definitely don't know anything about Integra AVRs but I would try setting static resolutions (1080P) for output of the BD player and AVR and input of the TV. Turn off any kind of HDMI handshake or control options and see if that makes any difference. Also make sure your TV's software / firmware is up to date if there is a way to upgrade it.
FWIW, I've had four different BD players (PS3, Panasonic, Samsung & Oppo) connected to two different AVRs (Sony ES and Pioneer Elite) to two different TVs (Pioneer and Panasonic) over the last several years and have never had any HDMI handshake issues with any of them.
If you can't get it working with and can't replace your TV I would suggest getting a BD player with dual HDMI outputs so you can get the lossless audio and 1080P video simultaneously. If you can't / don't want to do that at least use component video (Red / Blue / Green) and Digital (coaxial) or Optical Audio so you can get 1080i / 720P with 5.1 Surround. Or you could do component video directly to the TV and HDMI audio to the receiver if that's an option.
I'm not too familiar with Sony TVs and definitely don't know anything about Integra AVRs but I would try setting static resolutions (1080P) for output of the BD player and AVR and input of the TV. Turn off any kind of HDMI handshake or control options and see if that makes any difference. Also make sure your TV's software / firmware is up to date if there is a way to upgrade it.
FWIW, I've had four different BD players (PS3, Panasonic, Samsung & Oppo) connected to two different AVRs (Sony ES and Pioneer Elite) to two different TVs (Pioneer and Panasonic) over the last several years and have never had any HDMI handshake issues with any of them.
If you can't get it working with and can't replace your TV I would suggest getting a BD player with dual HDMI outputs so you can get the lossless audio and 1080P video simultaneously. If you can't / don't want to do that at least use component video (Red / Blue / Green) and Digital (coaxial) or Optical Audio so you can get 1080i / 720P with 5.1 Surround. Or you could do component video directly to the TV and HDMI audio to the receiver if that's an option.