I'm assuming you want to express that you don't have either. The meaning can in most cases be derived by logic.
"Wir haben kein A und kein B" means we have neither.
"Wir haben kein A oder kein B" means at least one is missing. You won't find such a statement outside a logic riddle.
"Wir haben kein A oder B" usually means "...
The word »aber« is a conjunction (»nebenordnende Konjunktion« in German) which means it connects two main clauses. So, let's use it as it should be used:
Unsere Gemeinde hat 5.000 Einwohner, aber wir haben kein Krankenhaus und kein Fitnessstudio.
Our community has 5,000 residents, but we have no hospital and no gym.
The part before »aber« is a full main ...
The explanation that you are giving for the difference between "aber" and "sondern" sounds very good to me as a native German speaker. What I can add is:
„nicht [a], sondern [b]”: means [a] is not the case, but [b] is the case. Contrary to [a]. (a sidenote: as a German speaker I find it weird somehow when other languages use the word ...