A suggestion: Give Joffrey a Slow trait. He's a child, yes, but his acts regarding Ned Stark, the riot of Kings Landing and several other things were more than foolish.
I think Joffrey being Arbitrary/Cruel suits his actions better. He's not dull witted, he's callous and spiteful - some of the things he did were stupid, yes, but I don't think he himself was dense, just what you get when you let a child who will eviscerate a cat to see if it has kittens in it have basically absolute power. I think Arbitrary is possibly the most fitting possible trait for Joff, actually.
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Also Nam makes a good point. The wildlings aren't rebelling, it's randomly spawning rebel armies that just keep popping up over and over again. Whilst it's realistic that it should be tough to hold beyond the wall, it's kind've ridiculous that several hundred men could just appear out of nowhere when none of the provinces in the north give any decent kind of levy - where the heck are they all coming from, effectively? You kill one bunch and another just reappears. (Also it's not mercs, they basically just pop up out of nowhere)
And then they take a province and just usurp it straight from you immediately. I know it should be tough, again... but the way things are currently designed it's almost impossible, which is kind've a let down if you want to stretch yourself and make it an objective to secure Beyond the Wall