...then you are looking in the wrong place. My local merchant charges about +20% for treated over untreated. e.g. https://www.cwberry.com/75mm-carcassing-timberThere is an option to buy treated timbers but they are twice the price.
Tudor times weren't built with treated timber and there are plenty of those still around
My wall is not perfectly plumb and therefore, it may be easier to build on the base and against the wall, to help deal with any variations.Could you not build it in sections in your garage and then bolt it all together outside.
I built a small shed like this and it was all framed clad and painted in sections
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