As you already have the DVI connector everything needed for HDMI is present. Only an mechanical adapter is needed.
I haven't researched, but I think the restriction is to use the name "hdmi" in marketing. Or mentioning it in documentation.
I think you could say "DVI with sound".
So time for that secret "dvi sound patch"
As for analog solution, I think the most viable ones has been outlined by now. Personally an op-amp that manages with plain +V rail seems neat.
Have you tested if the output works for line-in on an standard amplifier?