I don't think your employer is being reasonable at all in insisting that you cover scars. And what does he mean that they damage the company image?
Is he trying to say that the company only wants to be associated with "perfect" people? Which I think is worrying if that is the official company policy.
Given you work in sport, does he think that fit/healthy people don't have scars? Which is insane.
Does he imagine people will think you got the scars at work and the children might think rugby is dangerous? Again, ridiculous.
Or does he know your history and think that associating with a former self-harmer will damage the brand?
If the latter, he is an idiot. You are not just a former self-harmer. You are a survivor. His company should be proud to be associated with people that have been through dark times and come out the other side. Heck, perhaps your involvement with sport even helped that.
I think with children you are absolutely right to deflect conversations around your scars, but the reality is that if in future you do work with pre-teens/teenagers there is a significant chance some of those will be vulnerable, and seeing you as a person who survived and is living with scars that no longer define them could be a really valuable message to them.
The reality is people are not as good at hiding things as they think they are. People will always see. If people like yourself who have lived through trauma are forced to hide that fact and act ashamed of it, then others who are going through those challenges now will see that, and will also hide it, and feel ashamed. I don't think that is the best outcome for anyone. Except maybe your manager.