I have a problem. Maybe similar. 2003 Grand Caravan. If I don't drive it for 2 or 3 days, I turn the key and hear tick tick tick. It won't start until I boost the battery. Is additional current being drawn. I purchased a new battery and still have the problem. Could it be the starter, but this only occurs if I don't drive it for a couple days. I had a Viper alarm installed but I do not think that it draws significant additional current. I had the alarm guy check the wiring and stuff to see if it was drawing additional current or shorting. He says the alarm is fine and not causing any problems. I pretty much trust his opinion.
It is no a cold weather problem. It happens just the same when the weather is warm.
Any ideas.