Sorry, this strategy is still problematic. You talk about cutting Lee off in the North while defending Washington. Great idea, but how does Meade execute it? The only feasible solution to block Washington AND cut Lee’s escape route is to divide the army. By the end of June 28, Meade knew Lee was north of Hagerstown and marching in the direction of Pennsylvania. Meade chose the most sensible option: concentrating the army between Emmitsburg and Westminster on June 29. Meade’s greatest fear was that Lee had already or was in the process of crossing the Susquehanna. Meade’s thought process is focused on catching up with Lee, not cutting him off.
Suppose that Meade decided to do your strategy. He can’t use the main army because it would disobey the instructions to screen Washington. So what options does Meade have for a strike force? Meade's total strength is 112,735 men, Lee has about 75,000. However, Northern intelligence, the Bureau of Military Information, overcounted Lee's strength (a rare mistake for this new bureau) and so Meade would be concerned that Southerners would outnumber him if he divided his command. But suppose that he does so anyway. Given Meade's knowledge of Lee's dispositions, his best bet to block him would be Gettysburg, (the mountain passes at Fairfield if Meade wants to use the ridges behind Gettysburg), Waynesboro and Greencastle. The distances between Gettysburg and Waynesboro and Greencastle are 35km (21.7 miles) and 49km (30.4 miles). That's about two or three days of marching for the Army of the Potomac and on June 28 the Army of the Potomac is still far away from those points.
If Meade sends a strike force of, say, the V and VI Corps, (24,914 men) off to block those roads, Meade will find himself incapable of actually covering the line from Culp's Hill to Little Round Top. There will be no reserve whatsoever. If Sickles still obliterates himself by moving to the Peach Orchard, there will be no V Corps to plug the gap and there will be no VI Corps to plug up all the gaps in Meade's line and act as a reserve. And we haven't even considered what Lee is going to do.
By the night of June 28, Lee knew that the Union army had crossed the Potomac from Harrison (a spy) and sought to concentrate his army at Cashtown, near Gettysburg. Lee is bound to discover Meade's position near Gettysburg if Buford and Pettigrew still engage each other at Gettysburg on June 30 IOTL and he will eventually discover the blocking force at his rear. By waiting, Meade has ceded the initiative to Lee. Lee could strike a heavy blow on Meade at Gettysburg or turn around and defeat the blocking force before Meade can react. Cashtown is 13km (8 miles) from Gettysburg and 32-40km (19.8-24 miles) from Waynesboro and Greencastle respectively. You treat it as though Lee will just sit still while Meade traps him to the north. This would be a very deadly assumption. Lee is one of the most aggressive generals of the ACW, why wouldn't he decide to attack and defeat Meade in detail? A battle would be fought on Northern soil, whether Meade desired it or not. Halleck would see the risk of a defeat in detail and surely Lincoln would not have approved of such a dangerous strategy.