They are using "We have a dog." only with the meaning of "We own a dog." which last they are also accepting now. I know it is confusing, because it is possible for us to say "We have our dog.", but think of the other one in which we would not say "We own our dog." We are going to have to consider this as translating our idiom to theirs.
You need the "var" in any case otherwise "bizim bir köpeğimiz" means nothing it is an incomplete sentence. But you dont have to use "Bizim" since it is already implied in "köpeğimiz" (since it means our dog" so you can say either "Bizim bir köpeğimiz var" or "Bir köpeğimiz var" to express the same meaning.
The most literal (and acceptable) translation is being rejected. Consider:
POLİCEMAN: "Did you report your dog as stolen?"
US: "No sir. We have our dog."
There are numerous questions with this problem. While in some cases I might say, "I have a dog", not always.
If there's a Turkish distinction/alternative between the two cases, please clarify.
Why is it "köpeğ(i)miz"? Why is the (i) there when the same 'version' of "cat" does not have it (kedimiz)?
I understand that the 'g' mutated to 'ğ' because of the (i), I just don't get why the (i) is there in the first place.