"Barcelonian" has a long and respectable tradition, though, like Peter, I wouldn't use it. I think in my case this is because I associate it with troglodytes who think "vibrant" is still a vote-winner, and with Wikipedians, who may well work for the Ajuntament.
But what implicit contemporary demonyc rule does it break? Why is "Valencian" music but "Malagian" noise to our loopy little ears?
And what should it be? Barcelonan is much rarer and also old and lame.
My Martian euro-cent: we find Iberian coins registered in Barkino, and then there's the Libyan gangster Indycar Barca, and you have to admit that "We are Barking" is just sooo appropriate at the moment.