I wonder if that's really why boys were traditionally privately educated but girls not so much??!!
No, I think historically that was more to do with it not being "worth" educating women when all they were going to do was have babies!!!!! and not work!!!!!
My DS1 is also lively, sports loving and can be a bit of a clown but we went the tutor route for him and he responded very well to her - they "clicked" and he would do his homework for her without a murmur. I think that you will find that another year will make a big difference.