A few years back my son was in exact same height range as you describe. I took an old pair of his shoes and carved off the back sole. Then I cemented them to his new shoes, giving him an extra 1-1.5 inches in height:



Yes I colored the new heels so they would blend in. No one noticed anything different about them.

Too uncomfortable for him to wear around all day, so we kept these in the stroller and he would put them on when we walked up to the rides he wanted to go on.