Logitech Extend BYOD + Appliance Mode 4K Content Sharing

We have a request to configure some rooms to enable 4k content sharing directly to the screen/s, while keeping our Rally Bars in appliance mode.

In testing, the Rally Bar + Extend combo is great -- at 1080p. Because the Rally Bar can't output 2160p, we're bypassing the Rally Bar entirely in order to allow the Extend to output its full resolution to the displays.

I've got a distribution amplifier to test with, and the initial switch works great, but the Rally Bar starts fighting for control of the screens, causing random switches back and forth between sources before eventually settling on the Rally Bar output.

I'm hoping someone may have thoughts or ideas on this. I'm hoping I don't need to bring use a matrix switcher.