Presuming that the neck is too big you could try tacking/sewing in a tuck on the binding on both sides of the neck, make sure that the fold in the spare fabric is put in so that water cannot sit in the fold but is on the underside. I would suggest getting someone to help you, one of you on each side of the neck so that you can work out how much of a tuck to put in and measure it accurately, and whereabouts on the neck the tuck needs to be to get the right fit for the neck. I would only sew on the binding and not the fabric so that if needs be the tucks can be taken out and will not leave needle holes in the rug. You will need to use very strong thread that you cannot break with your hands.
He already almost outgrown it so I don't think it would be worth the investment. Also it's more that it's an odd shape than it's too big. It's fine top to bottom and it's almost too narrow at the shoulder now (it wasn't originally) but has a bulge in the middle that means it gapes half way down.