Such things are especially mega expensive.
That's why it immediately got dropped from the planning again
You extend the carport's roof over part of the house. This makes the roof bigger, but you don't have to build another roof.
But the carport will not be done by the construction company, rather privately by us in self-performance. If you do these "fiddly bits" through the construction company, in my guess the bill will end up just as high – for less living space. That's why I don't see the advantage. Besides, as I said, I don't see a huge saving potential on the upper floor, maybe 10 sqm. Also, that wouldn't automatically get rid of the "stair hatch" you hate...