Your problem is very likely the audio delay Sony adds to compensate for the inherent video delay. Unfortunately manufacturers like Sony, Panasonic, Toshiba, etc. "automatically" add an audio delay to not only their internal speakers but to their optical audio output as well to offset their TV's video delay. And unlike Samsung, Vizio, etc they don't give the user any option to turn it off.

As others have suggested sending the matrix output to the AVR "first" bypassing the TV might solve that issue BUT it "won't" if your equipment supports HDMI's auto lip-sync correction feature unless you turn OFF auto lip-sync correction.

That optional HDMI feature does not do what most think. It does NOT correct lip-sync! It has no idea how out of sync your signals may be. All it does is "automatically" add a fixed delay to cancel the fixed video delay of a TV. Almost all Av receivers have had that ability (to add a fixed delay) for at least 10 years so that feature simply does "automatically" what you could have done "one time" manually through an audio menu in perhaps a minute or two. It is a total joke as far as correcting lip-sync and in your case will deny access to the inherent video delay which might be able to correct your problem. It effectively would add back the audio delay you are trying to avoid.

Reason: The way it works (auto lip-sync correction) is like plug and play for computers: During the HDMI EDID session each sink device passes it's capabilities to the source devices and the source can only send a signal that ALL downstream devices can accept. Audio latency is one such parameter that "optionally" is exchanged if a device supports the feature.

Your Sony TV for example might report a video latency of 80 ms for progressive or 96 ms for interlaced signals and your AVR would automatically add 80 or 96 ms audio delay.

If that happens you will be right back where you started so you need to be sure auto lip-sync correction is turned OFF.

It won't work for your complex system with the matrix switcher but I normally recommend taking each source's audio directly to your AV receiver or sound bar to bypass the TV for two reasons: One, it will eliminate the TV's ability to delay the audio, and two, it will allow your AV receiver to receive surround sound which it otherwise won't get unless the TV supports Dolby pass through. Your Sony does but ironically most TV's that do support Dolby passthrough are the ones that add the audio delay you can't change.

Many users "think" their Samsung TV's which don't support Dolby passthrough can accept Dolby surround on its HDMI imputs but it will only be stereo due to the EDID negtitation in which the TV requests stereo . They are confused because the packaging and literature states "Dolby Digital" but that will only come from the TV's internal sources such as its tuner and streaming service and NOT its HDMI inputs. It is amazing how few users who actually own these TV's realize they aren't getting surround sound from their DVD or BluRay player if those HDMi sources go to the TV first and its optical output goes to their sound bar.

Anyway, good luck, resolving your problem. No company makes a consumer product to delay video. The companies who make lip-sync correction products such as Alchemy2, Felston, Primare etc. all delay digital audio via a hand held remote as the assumption is that your TV's video delay will be in the 50 to 100 ms range. That gives you a negative adjustment of that 50 to 100 ms inherent video delay but TV's like Sony, Panasonic, Toshiba deprive you of that negative adjustment by adding the audio delay even when you don't need it. Just like HDMI's auto lip-sync correction feature which again adds the delay requested through the EDID whether you need it or not which often makes lip-sync error worse than if left uncorrected.

I route audio from my HDMI sources directly to my Felston DD740 which has 4 inputs and on to my soundbar and "usually" the 80 ms video delay from my TV in conjunction with my DD740's 680 ms audio delay gives me enough adjustment to correct lip-sync. The range is -80 to +600 ms but even that might not correct cases where the signals arrive with over 80 ms audio delay already present.

Also some audio systems delay audio. Sonos about 30 ms which can usually be cancelled by the TV's inherent video delay but not if the manufacturer imposes the audio delay like you are seeing. Devialet speakers add over 150 ms audio delay and that can't be corrected unless you happen to have one of those 4K TV projectors that I'm told have over 300 ms video delay.

Ironically the manufacturers have worked hard to speed up their video processing and have reduced it from 100+ ms to as little as 40 or 50 ms in some cases but consumers are FAR better off with large video delays alloing them to adjust an audio delay since that can correct the rare cases where the audio arrives already delayed.