How straight is your son's straight bar?

If it has any rise at all that will limit how much you can narrow it. Trial fit your brake levers, shifters, handgrips, bar ends, computer, bell and anything else that you plan to mount on the handlebar. Move all of that junk in as far as possible. That's the most you can trim your handlebar.

For a frame of reference, a straight mountain bike handlebar is about 22" wide. That's 3" or 4" wider than a road bike bar. A mountain bike riser bar will be about 24" wide. The riser bar also sweeps back more and makes bar ends look goofy. I once tried cutting a mountain bike bar to match the width of my road bike bars and it felt too narrow. My advice is to cut about 3/4" from each end and try it that way for awhile. You can always cut more later but you can't add back.