So, you say that Jefferson Davis and Robert E. Lee hatched a scheme to invade the north and win recognition from England and/or France, and/or any other country. Is that how you'd go about it? Not likely.
To win recognition you must win a battle, take a city, occupy territory. The easy way to get noticed is to take a city. In this case Harrisburg would have been a real plum ripe for the taking. You can see the headlines, "Confederacy occupies Pennsylvania capital." Or "Lee takes Harrisburg - - - State assembly flees." Or many other headlines. All it would have taken was a day or two and move west to, say Pittsburgh and grab the steel mills, then into western Virginia (now West Virginia) and back home, with the Army of the Potomic panting along behind the whole way like a bunch of Keystone Cops (pun intended).
A fast move on Baltimore as another example might have gotten a quick and easy occupation, crippling the nation's capital by cutting it's communications and supply. All Lee had to do was keep moving, avoid major battle, and occupy a few major cities, strew destruction behind his army and make the Army of the Potomic and the federal government look incapable of confining the confederacy. Start thinking and you'll find many more examples of what could have been done with the Army of Northern Virginia to help gain recognition for the confederacy.
Instead, Lee did none of the above, and split the army in enemy territory, avoided a needed occupation (how would Meade have gotten him out of Harrisburg?), and finally threw the army away in totally inept generalship.
Yes, Lee was on the side of Union. How can one think otherwise.