Well, the idea that the Confederacy could ever have successfully captured Washington by storm or siege is utterly unrealistic. By 1862 Washington was likely one of the most heavily defended cities in the world, by 1864 it was the most heavily fortified city in the world. Any attempt to take it by storm would have resulted in wholesale slaughter akin to Grant's Cold Harbor battle, but in reverse, and likely much more one sided. Besieging it would have been impossible since it can be resupplied by sea if absolutely necessary and the Confederates have no way of stopping such an avenue of resupply even assuming they could encircle the city.

But I digress, your main point is about the idea of foreign intervention and assuming a hypothetical southern victory on northern soil. This would greatly depend on what year your talking. If 1862, it's possible, France did want to intervene on the side of the Confederates and intimated as much to the diplomats Davis sent. However, they were unwilling to act without a British declaration of war, and it is difficult to gauge how likely such an intervention ever was. Had they won Antietam, it is possible, but following that defeat they never had a realistic chance of achieving foreign support until victory for the Confederacy was certain and by that point what need would they have for such support.

You might be wondering why I don't include Gettysburg or anything following 1862 in my belief and that's rather simple. Had the Confederacy won at Gettysburg consider the situation on July 4, yes you've just won a "victory", but you've also lost 25,000 men, you're still outnumbered, and you can't possibly assail northern cities. To the north you're blocked by a river and the difficulties of crossing such a river. To the south you have only Washington and if you couldn't besiege it when you had 75,000 men you certainly can't now. You've lost a 1/3 of your army and remain deep in enemy territory, now what. Wait, hope the Union after 2 1/2 years just gives in, cause Lincoln wouldn't have and he was still president no matter public opinion. Oh, and about that, you're victory is about to be tempered by the fact that on the same day you won Vicksburg and some 35,000 Confederates just surrendered and the South out in the West is in full retreat. So the situation from your perspective, or even Union newspapers, hysterical at the loss and the mystique of Lee may not change. But in the eyes of the world you won a major battle and then lost a major battle and in doing so lost 60,000 men that you could not afford to lose. The public opinion in England was never high, why would it change now that you just went 1-1 in major battles, from their perspective all you did was just offset by what Grant just did and from a purely strategical sense to the military minds of England, Grant's victory was far more significant. Everyone in the South knew it too and Lee's loss, far from being the injury was merely the salt in the wound. It's also uncertain whether immediate intervention by the British or French would have caused an immediate peace settlement. It takes time to mobilize your forces and get ready for a war, the U.S. even after a loss at Gettysburg would have had a 6 month window before having to worry about British or French troops. In that time the South's fortunes out west got worse not better. Ultimately even as early as 62 such an idea was relatively unlikely, by 63 it was far fetched, and by 64 it was bordering on delusional.