Jon doesn't want the throne and I'm sure Littlefinger knows that. Jon just wants to stop the whitewalkers so I don't see why Jon would be an obstruction to Littlefinger. But like I said earlier, the "game" is nearing its end and Littlefinger has already lost, I think he's just trying to come to terms with that. He lost Cat, he lost Sansa and without Sansa he's not getting the throne and killing her brother won't help him nor did helping him. It's over for him. Wouldn't be surprised if he pulls a tommen
Littlefinger needs to get more leverage to claim the throne. He needs the North to march South.