Expressions like "It does a body good to get out in the fresh air," are a little old fashioned, and mean "It does a person good." But native speakers understand the expression.
The pun, of course, derives from the benefit to your anatomy, as well as your identity.. Yes, it sounds childish, but in a good way.
As a native speaker I would say it sounds a little awkward, but if someone said it I wouldn't think anything was wrong with it. I think it is fine grammatically; it's just seems a little clumsy somehow. It's mainly the 'a'. When talking about nutrition or exercise people usually say 'your body', not 'a body'.