Note that the Garmin 810 needs a smart phone to do the live reporting of position.

I suspect that using the Garmin 810 as the GPS for the positions the phone reports will use less phone battery than using the phone's GPS. I doubt the Garmin 810 transmitting the info via BT consumes much of the Garmin's battery either (ANT+ is similar to BT and the cadence/speed sensors last quite a while on a tiny battery).

If you don't use the phone for navigation, you might be able to use an external battery to allow you to do tracking for a longer ride (which would be much cheaper than a Garmin 810).