I just tried logcat, but there are no interesting messages related to HDMI. It's a really short output log, nothing like as detailed as what I expect dmesg to look like.
That said, unfortunately I was not able to replicate the navigation bar glitch again when plugging in the HDMI cable, so logcat wouldn't have picked up anything interesting anyway for me.
Disappointing, but still I'm hopeful something comes of this in a firmware update.