The Arcam SR250 is a 2.1 channel AVR with HDMI switching. They discontinued them a while back so finding one new may be a chore but they are pretty easy to find used. It has Dirac room correction and a decent internal amp but it may be on the anemic side if you have hard to drive speakers. I'm using one currently in the same situation you are describing and I really like it. I have mine paired with an external amp and it works very well for my use case.
You could also use your TV as the HDMI switcher and run an optical line back to any integrated with an internal DAC. I still do this so I can use the smart apps on my TV and run them through the main system.
The benefit with the Arcam is it can decode and downmix all the surround formats where a standard integrated won't be able to. You will have to change the output settings on the TV to send a digital signal to the DAC in a standard integrated amplifier that it can decode but it will still work.
You could also use your TV as the HDMI switcher and run an optical line back to any integrated with an internal DAC. I still do this so I can use the smart apps on my TV and run them through the main system.
The benefit with the Arcam is it can decode and downmix all the surround formats where a standard integrated won't be able to. You will have to change the output settings on the TV to send a digital signal to the DAC in a standard integrated amplifier that it can decode but it will still work.