You mean....the poodle exists???
Poodle is a political characterisation. I don't want to get into politics.
The fact is that there is much closer military, security and intelligence cooperation between the UK and US, than between any other pair of NATO countries. There are disagreements, but they are usually resolved without a huge public row, or an open dispute.
They also usually end up supporting each other's military and intelligence activities, even without a foreign policy decision by government heads - because the cooperation is so close - the Falklands War being a classic example. Even when the US government was still trying to be a neutral arbiter, UK forces were getting all sorts of war supplies from the US, and even getting Vulcan bomber spare parts from US military museums.
On the whole, Britain has benefited from this kind of cooperation much more than the US, since it needs the help more, and senior British politicians have generally historically been aware it - Blair certainly was - some people would say perhaps even too aware, and they would argue it that it was so much so that it blinded him to other considerations.
although it was clear that at least two permanent members were opposed to the Iraq war.
That was because Blair had politically boxed himself in at that point.
He figured that he could swing it (ignoring all the evidence and advice that he couldn't and shouldn't try), and he therefore committed himself to go for the 2nd resolution.
It would have been a lot more politically sensible for him for him to argue from the start that the 1st resolution and the Gulf War ceasefire was enough (which is of course what he subsequently argued), rather than too go for the 2nd resolution.
The US of course took the position that the 2nd resolution was unnecessary, and therefore wasn't prepared to do any real political arm-twisting or horse-trading to get it (which might have swung it for Blair if they had).
If it weren't for Blair, the US would have involved the UN even less. The US certainly wouldn't have bothered with trying to get a 2nd resolution even tabled at the Security Council, if it had not been for Blair wanting it. They would have just gone ahead without involving the UN at all.