You're right it doesn't constitute a majority of the country. But given that it was so large and was the largest ever, I reckon Blair could have given it as a reason to tell Bush we weren't doing it.
The US would have done it anyway. But Britain needn't have got involved.
Do you really think there were as many, let alone more, British people in favour of attacking Iraq as there were opposed to it?
It was all dodgy as fuck. The UN said no. There was the dodgy dossier and the dodgy death of scientist and biological weapons expert David Kelly (dies anybody actually believe he killed himself?!).
Blair could easily have said no. Imo, his who got the better of him. He'd been lauded as a hero and statesman for his intervention in Kosovo and I really believe he got a little bit intoxicated with power and wanted to be remembered as a great statesman.
In the end he's remembered by many as a war criminal