Alien Space Bats galore
Alien Space Bats (ASBs for short) is a neologism badly thought out plot devices used to create an alternative history
The first issue here is the Concert of Europe (formed in the after math of the Napoleonic wars to prevent a repeat of that mess) was having major headaches by the time of the US Civil War; the the Crimean War (1854–56) and the Italian War of Independence (1859) had just happened and it was clear the Austro-Prussian War was coming down the pike (it finally arrived 1866). So there was more then enough problems at home for France to worry about.
As for England there were alternative sources of cotton in India and Egypt and they were heavily dependent on food imports from the North. It certainly didn't help that the Confederacy had tried to blackmail England with a cotton embargo. As if that piece of stupidity wasn't enough the Confederacy then by all accounts sent people who were poor diplomats to the nations of Europe further alienating anyone even considering supporting them. It was so bad that European diplomatic missions were actually advising their resident subjects to refuse to serve in the Confederate army; the Confederacy finally got around to expelling them in late 1863 AFTER Gettysburg and Vicksburg.
Then you had the matter of millions of pounds in British securities that the North could have immediately seized.
In fact, the British Cabinet that met in October 1862 to discuss recognizing the Confederacy after the battle of Chancellorsville was so opposed to the idea that the three men who had call the meeting for this very purpose (Palmerston, Russell and Gladstone) didn't make any effort to promote recognizing the Confederacy. Another factor is the British Cabinet feared that when the Union won the war (a forgone conclusion in their book once the Union got a general that was reasonably competent in charge) that if they tried to support the Confederacy that Canada would be invaded by the victorious North and conquered in ONE short campaign if the Union got a reasonable general in charge (the Union actually got two that confirmed the British Cabinet's fears: Grant and Sherman)
Any meaningful support France had for the Confederacy had died in April 1862 with the capture of New Orleans and even before then Napoleon III was willing to help militarily ONLY if England also decided to help the South and there were concerns that sending troops would make war with Prussia more likely (this war in OTL occurred 1870 so France's fears were justified)
So it wasn't just slavery that kept British and French out of the US civil war but the Confederacy acting like arrogant jerks, fear of economic reprisals by the North and then military reprisals when (not if) they won the war by England, concerns about European matters by the French.
IMHO the ONLY ASB free option for a Confederacy "victory" would be for Lincoln to lose the 1864 election which could be easily accomplished by either delaying or preventing General Sherman's march to the Sea.