So, basically France joins in on Americas behalf because Taxation & Representation. Considering France isn't on top form, this seems odd. Especially considering that 1830 has the fall of the Bourbon Dynasty in the July Revolution.
Ignoring that as well as the fact that backing independence so didn't exactly go well for France last time - I'm skeptical, but I'll roll with it.
My main question is - why wouldn't other European Powers get involved against France and the US? Spain for example? The violation of the Monroe Doctrine, and Britain needing the help leaves Spain in a position where they can attempt to reestablish control over their colonies - which Britain can hardly complain about since they're doing the same - it may end up being post-war, but nonetheless, having the British OK will be useful for the Spanish, as they can take advantage of Mexican instability.
My other question is - why does American violate the Monroe Doctrine? After all the US agreed not to interfere with existing European colonies, or European affairs. Intervening on the behalf of the British colonies in the Caribbean is a violation of the Doctrine.
I can see this ending badly for the United States - mainly because with the Monroe Doctrine disregarded, and a window for Spain and Portugal to reassert control in Latin America - this could be one of the messiest wars in that time line. Fighting in Europe against France, bankrolled by Britain whose control over the seas isn't really challenged, with Spain and Portugal involved to boot.
Honestly, the biggest winners here? Indians. - The Boers might force the British to deploy Sepoys, or Indian Troops, or whatever name you want to give to an expeditionary force from India - in South Africa. A bloody affair, with the Boers outnumbered - and a change to British Military Doctrine to boot. Deploying Indians as traditional forces outside of India would raise their profile, and make White British be more fond of their Indian British 'brethren-in-arms'. (cue propaganda).
I'd say the USA suffers in such a war - if the Latin American states, especially Mexico, can organise during the war, a late entry could lead to them seizing a lot of the midwest, with Britain's aim being restoring as much of the Eastern Seaboard as possible. I think the US national psyche might well shatter if Britain lands Indian troops from the Boer Rebellion in America too. Indians beating on White folk? Might give some Amerindians ideas, especially if the British support them.
Yeah, American getting involved is a bad idea for America as it was at the time.