I've been paddling a Burn III (Medium) for a few months now. I'm 6'0" 135-140lbs, and I absolutely love this boat!
It looks like the weight range for the medium is about 115-175, with 145 then being the center. I think at your height and weight you could go either way, and that it will depend a bit on how you want it to behave, and how you intend to use it. If you want it to be more stable, but a bit less responsive, and want to use it for multi-days with lots of gear, go Large for sure. If you want a more playful and responsive ride for river running, go Medium. I remember reading a review from overseas talking about it being better to be above the center of the weight range to be able to fully engage the edges. I haven't notice a problem for me there, but when I paddle my wife's Karma S I do notice how much easier it is to get that boat on edge compared to my wider and flatter Burn. (That Karma S is like a race car!)
I met a guy at this year's North Fork Championships who was about your weight and was paddling an XL. He told me he had been in a Large first, but thought the XL was way better. He tried to convince me to go Large, but I think Medium is perfect for my size and intended use. To each his own!
"This river don't go to Aintry."