I think there was a misunderstanding. I wasn't saying the British didn't support the Monroe Doctine. I was saying they didn't support the Monroe Doctrine in order to allow the United States to conquer newly-independent Latin American countries for themselves.The British enforced the Monroe doctrine. Britain did it to keep its rivals out of the new world. Given the choice between a french buffer state and a larger United States, I would say the British would back its now ally and strong trade partner the USA over its long time rival France.
The British would not be happy with either a French puppet or US annexation even if they would be forced to tolerate one or the other.