ITTL the demographic and industrial advantage for the "north" versus the "south" is much larger than OTL. The slave states no longer have the population and resources of Texas to start with, and Texas as a free state as well as the free status of what was Northern Mexico means supplies from elsewhere to the CSA from Mexico and through the Gulf are not happening. Since the USN is going to be larger and more modern than OTL, the blockade of the "CSA" is going to be more effective sooner than OTL. This does not bode well for the "CSA".
On the international scene France will be leaning or actively "north" not "south". IMHO the UK while more "anti-American" than OTL is still not going to be that much more "pro-CSA". Yes, not having Canada means that concern for the UK is not there, and the resentment is there. However Nova Scotia/Newfoundland is vulnerable, so that goes in the mix. If the UK wants to support the CSA, more in the hope of weakening the USA as it fights the war and then recovers from it afterwards, their best move is to allow Drakia to be the conduit. At this point in time Drakia is still relatively tightly bound to the Empire, so if the UK wanted to remain neutral they could lean on the Draka enough to make any aid a trickle. They could, however, go "wink, wink, nod, nod" to Drakia and when the USA complains, as they most assuredly would about Draka blockade runners and aid, the UK can say that as a Dominion within the British Empire the Draka have freedom of action so that "private" actions like blockade runners with aid, and "sales" of supplies (on credit of course) is determined by them not London.
As long as the RN does not directly intervene, this fiction can be sustained. The Draka don't have much of a navy as the RN does the work, they could, quite legitimately, escort blockade runners through international waters with some naval force as a token, as long as the escorts don't enter US waters ("freedom of navigation" you know). Similarly they could outfit commerce raiders like the UK did, with the same fiction that they sold ships and guns but no "armed" ships (the Alabama met the guns in the Azores and fitted them there).
Of course a war is not inevitable. Rather than CS refugees going to Drakia after losing the ACW you could see Drakia offering substantial land grants to southerners, and free transportation and nothing would stop the southerners from bringing their slaves with them, at least early on. Only one in four southern households even owned one slave, and the number of large scale slave owners was relatively small although politically dominant. If a significant percentage of those slave owners decided to go to Drakia as described, having the brains to realize that secession was doomed then war talk peters out. It wasn't the yeoman farmers, however racist they might have been, who were clamoring for secession. OTL the option of selling land, and potentially some of the slaves and taking the cash and all/some slaves elsewhere with a large land grant simply was not viable. OTL Mexico was non-slave, and Brazil was not attractive until the south had lost. Here Drakia speaks English, is part of the British Empire which southerners find congenial, and converting slaves to serfs is a matter of paperwork unlike OTL where slavery was frankly outlawed in the Empire (in spite of a lot of slave like practices).
I expect the USA would be happy to see the major slave power folks leave, and even swallow the taking of slaves with them.
Just some thoughts.