If you want to keep it even
somewhat realistic, you need to go with a Trent War. Now, the very premise of a Trent War is very unlikely, because even if it broke out (which could
only happen if Lincoln failed to understand that,
yes, Britain
would declare war unless the British demands were met without delay or further mediation), Lincoln would apologise and sue for peace
at once. But let us assume that the conflict breaks out, and that patriotic fervour in the USA doesn't allow Lincoln to back down. In that case, you'd see something very much like
@Saphroneth's excellent timeline on the matter. Bottom line: the USA would be up shit creek, and no paddles would be forthcoming. Under the right circumstances, this could allow the CSA to get Southern California, New Mexico Territory, Missouri, Kentucky, West Virginia, Maryland and Washington DC. Possibly also Delaware.
That's by no means a likely scenario, and it can only ever be achieved if Britain and the USA are at war during the Civil War. This is truly the most far-reaching wank that the CSA could hope to get in any timeline that is not steered by the mighty wing-flap of an ASB.
As for screwing over the USA: the same scenario allows for that (beyond the losses already mentioned), because Britain could plausibly lop off some bits in the settlement of the war. Depending on how the conflict goes and how badly the Americans have pissed Britain off, you could
at the very most imagine a TL wherein Britain takes the area that would become Washington state, that little border area jutting north at Lake of the Woods, the Upper Peninsula of Michigan, Detroit and a modest border strip west of Detroit River (to secure British control of said river), a similar border strip east of the Niagara (to secure said area for Britain as well), a border strip along the New York side of the St. Lawrence (to secure control of
that river), and basically the northern half of Maine (to allow for a nice east-west railroad there, linking Montréal to Fredericton as directly as possible).
This, too, is a very extreme scenario, where US militiamen have by sheer 'luck' somehow managed to kill a leading British general in a dishonourable ambush or something like that, and Britain
really feels that the USA must be punished to the greatest extent. Under all other circumstances, Britain would certainly drop
some of the above claims.
Anyway, that's it, really.
Maybe, and this is rather out there, you could have William Walker's Nicaragua filibuster somehow succeed in a lasting way, and have
that be the POD. Then US fears of Nicaraguan pro-CSA blockade running would be amped up, and you could use that as the reason for a tough US stance against
all perceived 'foreign help' for the CSA-- resulting in an escalating Trent Crisis and a whipped-up public furor in the USA that stops Lincoln from just seeking peace. The results would then be the same as above, with the added fact that you could have Nicaragua join the CSA after the war, thus wanking the CSA just that little bit more.
I'm out. That's all I've got.