I share a similar view, that the CSA did exist as a political entity. That is, it built a funcitoning government, a political system, an army, a navy, etc. So, in my view, it existed as a country, albeit for a relatively short period of time.
I agree that the CSA was not formally recognised by international powers (particularly the ones that really mattered at the time) but for me I'm not sure how relevant this is to the topic at hand. What I mean by this is that if the CSA had won the war and gained its independence, international recognition would, very likely, have followed. I believe that formal recognition and support were not going to come unless the CSA won the war first.
If you wanted to be very technical about what the CSA was, it might be called a "region in revolt". But, I don't think this is immensely accurate as the political/national system the CSA built was quite sophisticated.
So in my opinion, for as long as it lasted, it was a nation.