Best way to get the British into the American Civil War..
Is have the French support the CSA
Can we widen this out?
French go all-in to protect their Mexican adventure and declare full support for the CSA.
And it has measurable effect.
At what point would the British intervene to support the North? Could this kick off another European war?
...apologies if this has been covered before...
The effect of the French 'declaring full support for the CSA' by which I presume you mean recognition of Confederate independence and material support of its armed forces would be to make the rest of the European nations including Great Britain give serious consideration to recognition of the CSA. Indeed if the French recognise them then the Belgians and Austro-Hungary would also recognise the CSA as they are all tied together by Royalty in to the Mexican adventure. Having said that in OTL the French Government made it very plain to the British Government via diplomatic channels that they wanted to recognise the CSA but were unwilling to do so unless the British did so too. Despite a strong desire within the British Government to end the suffering of the ACW they decided not to recognise the CSA or even offer to mediate between the beligerent parties.
With French recognition you get the interesting situation of the French Navy protecting French ships entering or leaving Confederate ports from the Union Navy. Perhaps even being ordered to lift the blocade by Force Majeure, something they were quite capable of doing but given the naval commitments to Mexico, Indo-China, The Magreb and home defence it would stretch them somewhat. However the Union Navy was far too weak and would not be able to offer a credible defence against such a move.
The British were less than thrilled by the French adventure in Mexico and its interests in Indo-China but they were unwilling to go to war about it. All they did was to refuse the French Government permission to use British merchant vessels as troop ships to transfer the army from France to Mexico. Instead they had to use disarmed battleships to carry the troops. This was ultimately one of the key factors in the withdraw of France from Mexico.