I’m not sure I understand this system at all. It seems as though you only have front right and front left ‘devices’, and no surround speakers at all. 
 

What happens if you disconnect that second set of four ohm speakers, which look like they are supposed to be surrounds, but are not? Does the Sonos Amp still cut out? I’m wondering if the Amp is shutting down to protect itself, from heat or other power issues. 

You are feeding an Amp rated for 4 Ohms per channel into 2 pairs of speakers that are 4 Ohms each for an overall impedance of 2 Ohms per channel, causing the Amp to shut down.  You need to disconnect one of the speaker pairs, change to 8 Ohm speakers, or install some impedance matching volume controls for the two speaker pairs. 

Or better yet, get a second Amp, connect it to the rear speakers, and then you will have true surround sound. 

AMP is rated for one pair of 4-Ohm speakers. You have two pairs connected. Speaker ratings are only approximate and impedance varies with frequency. AMP is protecting itself from the potentially dangerous load — as designed.

A speakers low impedance point is usually in the 60Hz range. If you are playing music with heavy low bass, protection tripping is likely. If you are playing music with minimal bass, protection will not usually be tripped.

Edit: some high end speakers are obnoxious and even though they are rated 4-Ohms there maybe a low point of 2-Ohms (or lower). These speakers will fluster most reasonably priced amplifiers. AMP is very tolerant of these speakers, but there are physical limits that must be respected.

Like others have said,impedance is too low. Connect your speakers in series and it will work properly.

Sonos warns that wiring the Amp in series is not recommended:

https://support.sonos.com/en-us/article/connect-four-speakers-to-your-amp-or-connect-amp

• We do not recommend wiring your speakers in series as it will result in poor sound quality. Refer to the image below to see an example of parallel wiring.

I agree that series speaker wiring will not result in the best audio quality.