Here's my thoughts. a HDMI header board will sit where the RF module was (or anywhere else you decide. a 3 pin input header on the board will connect to L, R, and common on the analog sound output. The 3 pin header will (hopefully) connect to 3 unused pins of the HDMI cable so the Analog audio signal is now travelling through the cable.
At the back of the TV a small header/breakout cable will allow you to use a small audio cable to the audio input and also feed the restored Video signal (I.E. without the audio) into the HDMI input.
This wouldn't interfere with your regular audio outputs. This obviously frees up the Audio cable at the back of the computer and keeps the cables tidier.