This is the only advantage to having a Labour rather than Conservative government in Britain. The conservatives would also oppose the new constitution -- a blatant bid for Franco-German hegenomy -- but would have less credibility with continentals. Blair has a better chance of rallying and leading continental opposition to this monstrosity.
Which, I might add, is one of the reasons for the Frogs' attrocious behavior. They were hoping to bring Blair down. He would either be replaced by a Labour leader more compliant to Froggie machinations, or by a Conservative government that would tend to turn its back on the continent.