Very nice game!
In general I think the problem with 2.Bc4 is not that the move is so bad it should already leave white worse (its not) but that its not good enough to be really critical. It doesnt really pressure black so he can just develop freely without worrying about some dangerous attack. Ive only encountered it in blitz where I usually go for an early Nc6,e6,Nf6,d5 gaining time on the bishop, but I think your idea is a fine idea as well.
You should be proud of that game, its hard to criticise any of your moves. I probably would have played Bb7 before f5 but thats about as minor a detail as humanly possible. Pat yourself on the back and thanks for sharing
I always hate to meet the newbie move 2.Bc4 against the Sicilian because while it should be objectively bad, the bishop becomes annoying when I try to get my d5 break in. In the following bullet game, however, I just played for counterplay on the wings and in the end I proved the bishop was missed on the kingside with a nice tactic.