Is "malformed base path in GetVolumeEntry" a cause of concern for Time Machine failing to start backing up?
I was unable to determine why the malformed base path in GetVolumeEntry is being logged, but I can confirm that it wasn't the cause of the hung backup where Time Machine was endlessly "Calculating Changes".

I stopped the backup in process and then used Finder to navigate to the folder where Time Machine stores the backups for this Mac.

Moving the 2012-08-24-032413.inProgress package to the trash allowed Time Machine to work properly (i.e. start backing up in less than 5 minutes, completing the backup in under 10 minutes). The odd message about the malformed base path remains something that gets logged at the start of each backup attempt, but it's not something that prevents a backup from being saved. 

In this case I cared more about a prompt backup than why the partial backup was causing the current one to take so long.