I have a hard time imaging a place they can go where the USA will not find them.
They wouldn't be hiding. They'd be in another country, presumably one that won't extradite them to the USA. There is no way that the USA is going to undertake military or police action in a European country to capture a Confederate government in exile. Such an act would be beyond the pale, and would result in war with the country in question, and possibly with other European countries-- if only to make sure that the USA will never even
think about pulling a stunt like that again.
And even if they do, I doubt they'd have much effect. The USA military is just too strong and too willing to use force (And the South too divided) for any guerrilla type movement to take off.
That last part is true, of course, but I repeat that there is
no way that the USA is going to undertake military or police action in a European country to capture a Confederate government in exile. So there would be a government in exile, and it would be utterly powerless.
In any case... I don't think any European country will be happy to host these guys. Nor would Europe be their logical choice. Most likely, they'd flee to Brazil. Because slavery is still legal there, and they like that. And because Brazil welcomed Confederate exiles in OTL. A lasting effect might be a bigger community of Confederados in Brazil, both because most of those who fled to other countries (such as Mexico etc.) in OTL might in this case 'rally to the flag' and follow the exiled government to Brazil, and because a number of those who stayed in the USA in OTL might instead by convinced to go into exile if such a move is led by what they perceive as their rightful government.