Logically speaking you are right. My car is not used daily. In fact it is not used for months sometimes. I start it and let it run for a 10 - 15 min to warm up. That is done maybe twice or three times a month. Under these circumstances expecting the battery to provide a full voltage of 13.5 is expecting too much. I have a battery charger which is used to charge the battery when it is unable to turn over the engine. To add to all this is the fact that the car has a dynamo instead of an alternator.
I have been fooling around with the battery, multimeter, clock and wiring over the last fey days. My conclusion is that the small coil inside the clock has gone weak and now needs a fully charged battery to keep it running. If the car was used daily, that would not be an issue.
One solution is to forget about the whole thing and start up the clock when the car is driven and another solution is to provide a 13.5V supply to the clock independent of the car battery :o
I have been mucking about with the second idea. I have made a small power supply see the pictures below:
Home made power supply has pins that allow removal from the car for battery replacement
Access to batteries by opening the box
I have tested the clock on this supply for over 24 hours and the results are very encouraging. A further improvement could be to use alkaline batteries that should last much longer. Over the last 24 hours I have not measured any appreciable voltage drop so my conclusion is that the power consumption is pretty small and voltage loss may be measurable after a few weeks.
Another thing that needs to be mentioned about this improvement is that it is completely reversible and in that way the originality has not been compromised.