Intervention in another country's civil war is a recipe
Intervention in another country's civil war is, historically, pretty much a recipe for disaster, and for the legitimate government to accept such assistance is pretty much a swift road to illegitimacy.
All conflicts between nation states are political, but civil war are, if anything, hyper-political, and inviting foreign intervention into a given "national" political question is a guarantee of sowing the wind, so to speak...
The US was willing to accept foreign volunteers are individuals (as witness GP Cluseret) but their commissions were USVs, and formed "volunteer" units were not something that were offered or sought, for obvious reasons.
The problems with "foreign legions" as formed bodies are demonstrated quite clearly by the British attempts at recruiting, organizing, and training the German, Swiss, and Italian legions for the 1854-56 war.
The long and the short of it is such units, whether volunteers or mercenaries, generate more problems than they are worth, and make it clear the cause and government recruiting them is weak.
Now, if the British had actually held to their professed policy of neutrality in terms of rebel purchases of arms and equipment from British sources in 1861-65, that would have been helpful, but there was too much money to be made by the good people of Bermuda, the Bahamas, and the Maritimes to shut it down...and in London, ultimately.
It took the threat of war by the US to get the British to move against the Laird company, for example.
Best,