For a 3 year old, any bike should work as has been discussed as the whole thing can work with no handlebar controls (ie, backpedal to break)
But a cheap solution to shifting as he gets older might be as simple as taking the shifter on his bad side and installing it upside down on his good side. You'll need to finagle it a bit to make it fit, but if you install it upside down the levers should be facing the right direction. (The numbers on the gear selector will be upside down....but win some lose some)
As long as he can hold the handlebars with his bad hand enough that he can loosen his grip with his good hand, that puts all the controls on the good side. Front and back shifters and at least 1 brake (could even put both brake levers on 1 side and zip tie them together so they function as one) all on 1 handlebar.
It's a a clunky sollution but it costs no extra money as every bike comes with all the parts. It would probably be impossible for an experienced rider who's spent 20 years riding with the controls shared on both sides, but if your kid grows up riding that way it will feel normal to him.