Wouldn't it probably be cheaper to simply build completely over the 12.49 x 12.49 than to saw around the base plate?
The best solution is to build over. Insulating is possible, but involves greater effort and is problematic with the floor structure and slope. The worst option is to cut open and insulate in between.
Didn't I already hint at that?
How about a conservatory?
Yes, that would clearly be the best option, I would say. You can do folding sliding doors there. A terrace in front would presumably already be outside the building window.