How do I convert him?
You don't. And you stop trying. You've already said you two have several points of agreement. Just stop there and let him support gun control. BUT...
Push the argument that what is needed, and ALL that is needed is enforcement (and punishment) of already existing laws. That's all we need. Just do the damn job.
See if you can get a commitment from him on that. THAT IS A VICTORY. He doesn't have to join the movement for repeal, or march or anything, except say STOP! ENOUGH! Do the job with what you have!
He doesn't feel the need for more than what he has, and like many, many unthinking individuals, feels that the rest out to be controlled, one way or another. After all, you can see 24/7 somebody getting shot on the screen, the news is so slanted a brick would roll on the slope. His world is his life, and what the screen and papers tell him it is. And absent to direct personal experience to the contrary, he'll believe that the answer is the one they keep telling him it is. Hell, see how he feels about Helmet Laws!? He may be a closet passionate libertarian on some other subject.
If he is, and actually uses logic for his basis on that subject, you have a chance of making him see the logic on the other subject. But don't force it. As long as you can get him to stop supporting then constant more!, More! MORE! from the gun banner crowd, as long as you can get him to see that "If I could get 51% of the votes, MR and Mrs America, turn them all in!" is NOT the way to handle the problem, its a win.