Actually there is a lot of overlap but I think of gym as mostly body conditioning and PE as encompassing much more, like balance and dexterity.

Lately we've been using something called a Trikke which requires a lot of mental effort to make work, like roller blading or ice-skating, except I think it's more whole-body.

Also, Yoga and stretching come to mind.

As to how to fit it into the schedule, I can't help you there except to point out that it might be easy to fit it in where there's room, evenings weekends or even as a reward for completing other more academic work...