I may be a little late but I feel the urge to share my thoughts anyway. (btw: I apologize for some grammatical errors etc. English is not my first language!)

I personally think "Laura" is about a guy who is secretly gay or knows that he has homosexual tendencies.
He asks Laura to marry him (music video, the proposal scene) so he can convince the world or/and himself that he is straight ("Won´t you just tell Cincinnati/I´m gonna need your love").
The "band" that he "knows" he´s "in" has not to refer to The Scissor Sisters. It could also be a metaphor for his true sexuality, the gay "band" that he knows he´s in.
Babby daddy could be a reference for his band mate "Babydaddy" who is also openly gay.

Well, I have to admit that the hairdresser-themed bit does not make much sense in this context.
It could be the place where he and "Laura" met but I honestly think it has no important meaning in relation to the rest of the song (or I just don´t get it.).
Songs that deal with homosexuality have the tendency to be a bit wishy-washy (for example Adam Lambert´s "What Do You Want From Me". He does not clearly state that he addresses a man but, somehow, you get the impression that he DOES or at least you wonder why he is so god damn cryptic.) (It´s still America, y´all...)
Also, if you pay attention to the music video (of "Laura") you recognize that the male character acts kinda weird (dismissive, that is) towards the female lead.
Anyway, these are just my impressions. I could be wrong, still. But I hope I could make a point. ;-)