Seems that after you start your car in colder temps the battery is reporting low voltage Hmm wait, no… the batt light only comes on when the alternator stops putting out voltage I believe. Some systems report low voltage with the battery light and some or most report on the condition of the “State of charge” of the alternator… A vehicle of this age is more than likely reporting on the alternator…which does need proper voltage from the battery to put out proper DC current to charge the batt.
I would begin with checking your batt connections. They must BE clean and tight…no corrosion or loose clamps… If it is a side post battery (The most hated style personally) the same applies. I would for giggles simply clean the connectors and posts and retighten everything and see how it goes. You can also monitor the battery with a voltmeter…esp when the light comes on…note the voltage. On the hated side post batteries…sometimes the plastic shrouds that encompass the batt cable ends interferes with or gets in the way of being able to tighten the cable to battery… Many times I use a razor blade to trim this outer plastic casing to allow the cable end to properly contact the battery…but thats just me and I see a lot of battery connections in my day to day.
It sounds like your batt is just able to operate…and after a start then hits a voltage that is too low to properly operate the alternator… The batt connections is the very first place to start. A volt meter wouldn’t hurt anyone either. Let us know what you find.
Time to run…I need to complete repairs on a Rolls Royce Silver Wraith II in NJ… Fun fun fun…