I would get him the model with screw in chokes. You will likely find the "turkey" chokes a bit tight for trap, esp the beginner.
As your finding out trap is a full choke game. Birds flush for 16yrds in front of you and depending on your speed they get out another 15 yards or so before you fire.
For informal, learning and just plain enjoying the sport of trap, YES you and your son will be just fine with a handi.
BUT...having said that. Shot-shell shotguns are a different animal than a rifle or a slug gun and my only real concern is fit.... As you point a shotgun not aim one it needs to fit to be shot well. You need to be able to reliably and consistently shoulder the gun and have everything properly lining up to consistently break the birds. Thankfully H&R's are pretty forgiving, easily lending them selves to stock shortening.
CW