Would suggest something is causing a reasonably quick drain on the battery, or as you suggest battery is on its way out but this should not set off the alarm unless the battery is totally flat.The alarm system ought to have a battery back up, though I seem to remember this was an option at one time but being CSW's could have been standard.
My battery monitor still shows in the yellow after a week of rest.
I didn't start my 110 from christmas to February this winter due to my broken ankle and it started first time. It was parked outside in the snow all the time alongside my Honda Civic which was completely dead.