I think the real question to ask is if what the additional rec provides is critical. A rec from a coach for an athletic recruit could help. A ref from a friend, probably not. A math teacher might know a kid is good at math. A coach for math competition might have additional, helpful insights.
You also run the risk of showing that you are going to be “that parent”. The one who, in spite of instructions, feels entitled/compelled to do it their way. The one to request a different roommate, to arrive outside the scheduled window, to want private lessons from a teacher not affiliated with the school, to tell the coach how to play their kid, etc. I’m not saying you are that person, but these schools have met lots of them and don’t want them. So while I fully understand the desire to provide as,much glowing information about your kid as possible, I would proceed with extreme caution and discretion.