The issues of whether a child should or shouldn’t take a shower during the nine days when his heart is broken and he is going OTD is mute. It is not a decision or a choice a parent makes for the child, it is a choice that child is making for themselves and therefore it has nothing to do with the parent. Hashem does not judge the actions of the choleh in such a case and therefore neither the parents, nor those peeking in on this situation should either judge nor discuss whether he is permitted to do so. The achrius is on the child alone and not the rest of the family. If the child was ill in any other way there would be no question, if he had a fever he needed to break, if he was sweaty and felt his bones were aching, etc. there would be no question and there is no question here. He is doing what he is choosing to do under the pain and weight he is burdened with and the parent has no control right now over his choices.
There is no point in picking apart his choices and what he is doing, it will continue to go against the norm and against the grain. What one needs to concentrate on is their own actions, their own tefillos and their own histadlus. In order to get through this most difficult nisayon one truly needs to strengthen their emunah and bitachon and one does that by saying “Hashem I know that you are listening and I know that you are watching over my child why he is going through this nisayon, please keep him safe and let him know that you love him and that you are by his side and will never leave him.” That is the most important part of this whole journey.