First question is if you are on the latest version of Zwift?
I seem to recall there were some cases reported about a year ago where users would see the red light and avatar would not move and resetting the RunPod (by removing and replacing the battery) would solve the issue. Zwift released a fix for this issue so that it would not be necessary to reset the RunPod to fix this issue. You should not be having this issue with the latest version of Zwift (or else maybe the issue has returned somehow).
As a test, next time this happens, you could try replacing the battery with (and I know this sounds weird) the exact same battery you just removed. Because this was the workaround for the previous old issue. Removing and replacing the battery effectively resets the RunPod if it has been placed into a weird state. If removing and replacing with the same battery still does not work, then it sounds like the issue is really about battery drainage.
You say the RunPod has been idle on your shoes, but are the shoes idle?
After re-reading your message, I get the idea that you only wear the shoes for running, but is there anything in the environment that could cause vibration or movement to the shoes. You can test for yourself that a small tap on the RunPod will wake it up. So, and I know this is not the case, but if the shoes were on a washing machine, the vibration would continually wake up the RunPod. Again, I know that is not the case, but it may be worthwhile to see if there could be any vibration or movement around the RunPod that is causing it to activate when you are not actually running on the treadmill.