Not with your equipment, and even with other Sonos devices with an analog line in, it wouldn’t work. Simply stated, Sonos isn’t designed to be a PA/Karaoke system. There’s no way to sync an analog input to a video input. 

I actually got this to work pretty well. I purchased an HDMI extractor to split the audio and video coming from the HDMI output on the AppleTV. Video routed directly to the TV and audio routed from 3.5mm jack to RCA to a new Yamaha sound mixer. The mixer also received microphone inputs, mixed with the AppleTV audio and then routed via RCA to my Sonos Connect to broadcast to my other Sonos speakers. Free karaoke videos/playlists from Youtube can be controlled via iPad/iPhone and AirPlay to the AppleTV. Works great!

I’m surprised you aren’t getting a significant delay with the audio going to the Connect.  It has a buffer in it which will delay the audio.

I got the mixer below about a month back. I’m not using it with a TV or Sonos at all, but as I understand the features, it can take the input from your Apple TV, mix in the mic audio, and send via HDMI to your TV (and thus to your Beam via HDMI-ARC.  I may try and test this out over the weekend.  The one downside I see is that the mixer is only stereo, so it might not be acceptable as a permanently in place system.

There’s only the slightest delay, and it’s barely noticeable. Since any AppleTV (karaoke videos) audio is synced with the mic audio through the mixer, it matches up nicely. The tiniest of delays between me actually speaking into the mic and the audio amplified through the system is consistent with what I’ve experienced in other karaoke settings, and adding a little reverb/karaoke echo effect on the mixer mellows it out.

I haven’t tried routing the audio from the mixer back to the TV to the Sonos Beam as you suggested because couldn’t figure out how to re-join the audio and video after first extracting. I imagine I could route mixed audio to the TV to play through my Sonos system in the same way it plays TV audio, but my issue was figuring out how to also pair that audio with video if I needed to extract the two to mix.

How can I do this with a Behringer mixer 1202, sonos arc, amp and Sony TV? I have got Hotec wireless mic and want to use YouTube on my smart TV to play the videos.