All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
The matter is much simpler than that. If your child touches the shower with his hand or touches some parts of his body, then this does not necessarily mean that they are impure (najis) unless you are certain that he touched impurity with his hands.
So you should wash your child normally, starting by removing the impurity from him and then washing the parts of his body that were stained with impurity.
It does not harm if he touches you with his hands, touches his body and then you, or touches the shower and then touches you, and so forth, as long as you do not know with certainty that his hands are impure.
Perhaps you suffer some sort of waswasah (obsessive whisperings) in this regards, so you must avoid these whisperings and not pay any attention to them and act normally and not rule that something has become impure except with absolute certainty.
Some scholars are of the view that a small amount of impurity is totally pardoned, so you may act according to this concession if you are affected with waswasah.
Allaah knows best.