Should I use the possessive in a sentence like the following:
1) I dine at Alive Cafe (KFC).
2) I dine at Alive Cafe's (KFC's).
Is there a rule about the use of the possessive case here?
If I were a native speaker of English, I would never shut up. :-)
No. Only use a possessive if the eating establishment has a possessive in its name, or is known by its owner's name.
You would not say that you ate at 'Alive Cafe's" but you could say that you ate at Joe's (Restaurant/Cafe/etc.)
(As far as KFC goes, some people used to call it "The Colonel's." Probably they still do.)