Sorry, not knowledgeable enough to speak to the base mute question, but a thought for a part-time work around. I'm on limited basic cable, and in the 900s there are numerous music channels. I'd think you have them as well. Check around for something that suits your taste, lower the volume, and have background music for at least part of the time you'd normally mute. There is a screen display of the channel, artist, and album, so it wouldn't black out the screen.
As to an alternate input with no source, can't see how that would create an issue, other than having the gear on all the time and the subsequent heat issues. Depending on access you could always pull the power on the cable box, or the output coax to your receiver when not watching ? Of course pulling the power will require the box to re-sync when plugged in again. Good luck finding the solution that works for you.