What should I do with my seven-year-old son who is prevented from praying at the Sharia institute on the grounds of getting his clothes wet, given that my husband believes he should pray at home instead of creating a problem?

There is no objection to delaying the Dhuhr prayer for a child until he returns home, because prayer is not obligatory for a child; rather, he is commanded to perform it as a form of training. Delaying the prayer due to need does not contradict training and raising him to revere its importance.