The sentence goes like this:
ハジメの場合、単に面倒というだけでなく、学校の居心地がすこぶる付きで悪いが故の憂鬱さが多分に含まれていたが。
I know the sentence goes something like this
In Hajime case, it wasn't just mere trouble, the reason he was not feeling good at school was due to feeling depressed.
But I still don't get the で in 付きで悪い
It's not 付きで悪い but 居心地が悪い ("to feel uncomfortable") modified by すこぶる付きで. すこぶる付き is an uncommon emphatic version of すこぶる (although the former is a no-adjective and the latter is a standalone adverb). This ～付き is a suffix that means "with ～ added" or "which comes with ～", so the gist is "it's so bad to the point where すこぶる can be added". BCCWJ has 7 entries of すこぶる付き/すこぶるつき, most of which are used by elder writers.
