Integrating RME ADI-2-DAC FS based 2 channel system into surround system?

Find a DAC with analog inputs, then you can just run the front L/R preouts from the AVR (if it has them) into the DAC and you’ll be all set with no selector box necessary.  Something like a used Bel Canto 2.7 would be excellent, and I can attest the preamp section is high quality and very transparent so would work very well in your situation.  Just another way to go in case it could work for you.  Best of luck. 

I think one option that doesn't increase your cost would be to run the ADI-2 into the analog inputs on your receiver, and switch your receiver into pure direct mode.

I think a better analog option might be just having 2 cables going to the power amp via a selector box or just swapping connections on the amp to keep the hifi side as pure as possible.  Probably over thinking everything as usual.

If you split the audio before the receiver, you're going to have a problem adjusting volume because you don't have a global volume control anymore.

You could look for a receiver that has a digital out option for the two L and R channels, but I think that may be looking at a price increase well above $800.

Even with the pro version, I think you would need to manually set the volume whenever you are switching to HT use in order to level-match what you calibrated on your receiver. And then remember to undo that when switching back to 2-channel listening in order to make sure you don't accidentally blow out your eardrums.

I think one option that doesn't increase your cost would be to run the ADI-2 into the analog inputs on your receiver, and switch your receiver into pure direct mode.