I don’t think this will work, at least the way you’ve explained it. But the Sonos devices, which should be connected to each TV, should play whatever is on that TV. Which could be the same thing, or something different.

FWIW, I was never able to get two TVs, both connected to DirecTV boxes, to synchronize while streaming the same channel. 

I get what you’re trying to accomplish but I don’t think your plan is exactly right.

I would look for a hdmi matrix with at least 2 inputs and 2 outputs, as well  separate optical output, such as this.  The idea is that all of your sources are connected to the matrix, both TVs are outputs to the matrix, and you use the matrix remote to play different or the same source on the TV.  The matrix I linked to allows you to select whether you want the audio from TV A or TV B.

I would plan on placing your sources and matrix somewhere near the playbar, as optical cables can’t be too long.  Probably next to your living room TV.  That leaves you with 2 other challenges.  First, you’ll need to have an HDMI cable between your matrix and kitchen TV.  I was able to do this as I have a one story home with attic space I could easily access to wire a 30 ft cable.  The other challenge is going to be control. The matrix remote is IR, so it needs line of sight, which could be a problem in your kitchen?  You would also have remote(s) for your sources.  All of my sources have bluetooth remotes, so that’s not a problem.

I have a similar setup, although there are more TVs involved, and the TV in the kitchen is really just a dumb computer monitor that always plays in sync with the main TV.  I found this to actually be better than a TV because I did not want  the extra inputs, ‘smarts’, or built in speaker on the TV, and it automatically goes on/off when a source is detected.

edit:  Forgot to mention.  It’s important to consider video quality when using an HDMI matrix/switch/splitter and multiple TVs.  If your TVs are capable of different video quality, many of these devices will downgrade to the lowest common denominator.  Others will want you to manually set the video quality.  Others will pick a higher quality and be able to downgrade the signal to TVs that can’t handle that quality.

How would you connect both TV’s via HDMI? Most TV’s do not have an “out” for images, so this would be impossible?

If the TV signal is coming from an external device, like a Cable Tuner box you could split the HDMI there to both TVs.

How well sound would sync between the TVs would be up to the TVs, Sonos should play what it is sent from the local TV without problems.