HDMI Audio Only Capability?

In order to use my Thunderbolt to listen to Internet radio (Pandora, Slacker, etc.) and my stored MP3's through my car speakers I had to purchase an auxiliary audio input device named Blitzsafe.  Rather than use the headphone jack on the Thunderbolt I was wondering if the HDMI port could be used instead?

I also have a Blitzsafe aux adapter installed in my truck. I purchased it mostly for the iPod connection. In addition to the iPod connector the Blitzsafe has the 1/8" jack and also RCA line level inputs. I wanted Bluetooth music. So, what I did was purchase a Motorola T605 Bluetooth music car kit (was at the time $32 on Amazon). This added Bluetooth music streaming plus handsfree calling to my existing Ford system. The music streaming goes into Blitzsafe adapter AUX3. It can be installed using just the music portion. Sound quality is very good being the T605 is hard wired through the Blitzsafe adapter RCA inputs.