The shoulder seasons (spring and fall) are the best times to tent camp at the Fort due to the mild weather. Summer is hot and wet, winter can be cold and unpredictable.
The spring campsite rates (from mid-Feb through Easter) are pretty high. From Easter to early June rates are lower and weather is warmer and the Fort is not as crowded. However a lot of people come in to camp just for the weekends which can make getting a ressie tough to get.
The Disney ressie system is designed to discourage short stays and encourage long stays. I checked arriving May 5, 2013 and leaving May 31, 2013 and got availability at all campsite levels (Partial, Full, Preferred, Premium). The max ressie length is 30 days. So the trick we all use if needed is to book as long as you have to in order to get the dates you want, pay the one night fee for a deposit, then call back in a few days and drop the unwanted dates. It's silly but that's the game you have to play.
So get that reservation made then check with us closer to time with any questions you might have. A tent would go nicely on a Partial, Full, or Preferred but which one depends on the size of the sand pad you want/need, general location of the loop, and how much bling you want to spend.
Bama ED
PS - Plus if you will be at CSR in October, you might want to take an afternoon to visit the Fort, check out the various loops, maybe eat at Trails End (we get the Giddy Up & Go (GAG) take out dinner of fried chicken and sides for $25 which feeds 4 people - they have picnic tables on the porch to sit at), and stay for the campfire/singalong (up to you if you want to stay for the movie, heh heh). A pre-visit will better prepare you for your real spring adventure. And if the GAG would be too much food, they sell the Lone Ranger take out chicken meal (serves 1 person) for about $10 each.