There hasn't been any reports on that dongle being required to use the digital optical connection with a HDMI connection. Considering that the dongle isn't available for purchase separately(at any price), I would assume it's not required and it was only included in the Elite SKU as a convienience.
Tommy McClain
The dongle is currently on sale seperately (Available here for instance). And while I've yet to see see any major websites reporting on the incompatibility between the 360 AV cable and the HDMI cable, I've seeen numerous postings from people on message boards (Like AVS Forum) stating that the two MS cables cannot fit into the console at the same time. Some people have reported that slimmer third party AV cables can be plugged into the console at the same time. However, it appears that the video output doesn't work correctly when you do this, meaning the AV cable somehow competes with the HDMI cable from a video output perspective.
Finally, all the MS documentation refers exclusively to using the new audio dongle to output sound via TOSLINK when used with the HDMI cable, as can be seen here.
Given that MS saw fit to include the new dongle cable with the Elites in addition to the full AV pack, I'm reasonably confident that this is now the only solution you can use to obtain sound when using an HDMI connection. I don't have a 360 myself yet however, so I can't confirm whether this is the case or not.