I think the greatest challenge is to remove the overconfidence felt in both London and Richmond at the time of intervention in January 1862, both seemed to think that the Royal Navy was an instant win button. Though it destroyed the Union merchant fleet and opened the Confederacy to economic freedom, it underestimated both the depth of Union resolve, and the unlike the CSA (which had no navy) and Russia (which had one but declined the fight) the United States was more than willing to fight on the seas and in the coasts and on the rivers.
For that matter the Confederate campaigns of 1862 were total slogs which cost far more casualties than the Confederacy could hope to sustain, while the British pinprick attacks on the coast distracted tens of thousands of Union troops, their numerous attempts at invading New York never panned out. By 1863 The Confederacy was reaching the apex of its military power, while the Union was only beginning to gear up. Britain was seeing just how costly this war would be. With a stalemate in the north it was only natural the British would want to bow out gracefully, and then the Union could turn over 100,000 more men on the Confederacy, leading to the final surrender of Confederate authority in January 1866 at Houston after for all intents and purposes the Confederate government ceased to exist when Atlanta fell in September 1865.
If you want the Union to lose have the invasion of Canada go poorly. The Union thrust up through the St. Lawrence could have easily gone the other way, but with Charles Smith leading the army (and willing to engage in risky maneuvers to reach and besiege Kingston) once he appeared in the British rear at Montreal, the British had no choice but to fall back to the island and wait things out there. If the Union army is busy trying to reach Kingston the British might actually pull off their invasion in 1863 and march as far south as Albany, leaving thousands of Union troops distracted into 1864 when the campaign to take Richmond bore fruit, meaning that the Confederates could probably hold on the Rappahannock and Lincoln has an even tougher uphill battle to get reelected, and maybe the Peace Democrats win leading to negotiations in 1865.
Or you could have France work more overtly in favor of the Confederacy. Maybe if they weren't mired in Mexico they would have jumped in rather than just being neutral.
However, if you keep Union troops distracted up north in 1864 I'm willing to bet that Lincoln won't win the election.