I think you're going to be fine, mostly.
You may or may not face some trouble reaching Khanewal (I'm not sure; never been to Multan!) but once you get there; The N-5 Highway will lead you straight to Lahore without any trouble.
Expect some trouble from Thokar Bypass onward, unless you head straight towards either Multan Road or Canal Bank Road which are almost always open.
By taking these two roads; you can get pretty much anywhere in Lahore with minimal trouble.
Drive safe!