Often questions like this (birthday) are easy to decide, not in this case. I did a search on German language websites for the words Beethoven AND 16. Dezember: about 280 sites have both entries. I did the same for 17. Dezember: 300 entries. Too close to call! One site had both dates and said: born presumably December, 16th, baptised December, 17th. Let's assume he was born during the night and there was no clock in the house.

Wolfgang