Hi, I am an American Board Certified Neurologist with experience in treating all neurological and all other medical conditions including the concerns of risk of viral spread from your child you have mentioned here and I am very glad to help you with your health concerns.
If he is just going to the swimming club but he himself is not swimming then it should be okay
Also I see that he is not coughing now
It is certainly safe now for him to be around with other kids and other people
The gastrointestinal viral will spread through oro-fecal route (through saliva, vomitus, fecal matter/stool etc) or sometimes also by coughing etc, but in his case these do not appear to be the issues because he himself is not going to swim so the virus is not going to enter the water and spread to others, also some gastrointestinal viruses can spread through coughing too but in his case it seems he is not coughing anymore so looking from any angles he is not contagious now. Through food & drinks too gastrointestinal virus can spread so he should avoid sharing these with others.
Once again I reassure you that your son is not contagious to others as long as he himself is not participating in swimming, also if he is not sharing his food, beverages etc to others, and also if he is not coughing anymore, so he can certainly visit the indoor swim club as scheduled tomorrow without being contagious to others.
Please let me know if you have any further questions, and if you feel you have received the answer to your question please rate me to credit for the time and effort put by me in answering your question.