So, after a soft reset and a full discharge/recharge, I have some progress to report.
I left my garmin on my desk overnight, running. It wasn't taking HRM readings and it was static for the entire time, but I'm guessing some of this saved power was possibly offset by having to hunt around for GPS signals as I'm miles from any windows.
I came in this-morning, watch fully discharged. Charged up again, checked history. It ran for 12 hrs and 21 minutes.
I thought that was pretty good, and it probably means I don;t need back-up power.