I find most children to be "free and playful" by nature, no matter what they are doing and where they are doing it.
I gather that we have a philosophical difference on how we intrepret "free and playful." I think that you are walking a fine line between "frenetic activity" and getting a paintable picture.
By all means try the beach...if you can find someone run after this child with a large reflector.
Or, maybe the parents could hire a professional photographer to work under your direction to get a decent shot so that you can paint a decent portrait?
I am curious...could you post a pix of a painted portrait - any portrait that you really like of a child, where there is movement, freedom, playfulness and "fresh-looking" clothing? I assure you that I am not boing sarcastic here...I really do not know what you envision your painting to be like. Here is the URL to a link with lots of professionally painted portraits of "girls outdoors":
http://www.portraitartist.com/girlsoutdoors.htm Do any of these portraits have all the elements that you wish to portray?
We use so many words to describe visual things and these words mean different things to different people. It gets complicated sometimes...