You are breaking down the sentence incorrectly. Think of it this way: "He resumed packing. While he was packing, he had a hard knot in the pit of his stomach." I think you are thinking of the hard knot in the pit of the stomach as something you're using to do the packing. It isn't, and the reason it isn't is a matter of context.
"With something" can mean "by using something," as in "I hit the ball with my tennis racket," but it can also mean "accompanied by something," as in "I went to the store with money in my pocket." You don't use the money to get to the store; it just goes along for the ride. Harry's stomach knot is an example of this latter case.
If, for example, "he resumed packing with his special magic packing spell," then he would be actually using the spell to do the packing. On the other hand, if he "resumed packing with a weary reluctance," his weary reluctance simply accompanies the packing. It is this way "with a hard knot in the pit of his stomach" as well.
So, again, it is a matter of context. You can't use a knot in the pit of your stomach to pack a bag, so if you resume packing with a knot in the pit of your stomach, you have that knot and you are also packing the bag, all at once.
The sentence that immediately precedes the ones you have provided is "They packed up their things, Hermione dawdling." (Got to love google!) So, when Harry resumes packing, he goes back to packing his things, and while he is doing that he has a hard knot in the pit of his stomach as well.