I’m not sure that a Confederate victory at Gettysburg would have meant Southern independence. Meade’s army might have suffered a defeat, but it would still have been largely intact. Far from losing the support of the Northern population, I think that a successful invasion of the North might well have galvanized support. Keep in mind that those in the North who opposed the war did so because they felt that it was not moral to fight against fellow countrymen who were not truly a threat. New York City was a hotbed of such opposition. Do you really think if Lee had threatened to attack NYC that opposition to the war would have continued?
The war was won by the North for the very simple reason that the Civil War was really the first industrial war, and the North far outpaced the South in terms of industrial capability. Think of how the war actually was won. It was not won on the basis of any improved Union strategies or tactics. It was only won when the Union finally put a general in charge who ignored the heavy casualties and stopped targeting Southern cities and territory and focused on the real task, destroying Lee’s army. That was only doable because of the immense advantage the North had in terms of manpower and industrial capability. I truly don’t believe that there was any way the Confederacy could have defeated the Union militarily given the limitations that the Confederacy had.
Had Lee won at Gettysburg he would have been in position to attack DC. That threat alone may have prompted the Union to negotiate a peace treaty. A defeat in DC almost certainly would have resulted in a final peace, or at least a years long peace between wars.