I don't think it would be a matter of the American Club hiring members only, but more so that a "visitor" cannot work in Singapore while "visiting". If the teenager is here are a tourist visa or a LTSVP, he won't be able to work. If he comes over on a Dependant's Pass, he would probably be able to get a summer job. So, it's the immgration status that matters.
In any case, look into the local Community Centers for classes. Although, be sure it is classes suited to his age group. (I'm sure he wouldn't want to be hanging out with a bunch of "old" people!) There should be some afternoon classes etc in different types of marshal arts, dance, etc. Golf is a good idea. So, is the Bukit Timah Saddle Club. A person can enroll for a set of lessons (4- once a week) Begginer's Course without having to pay the Club fees. There is also volunteer work at Riding for the Disable (RDA) and the SPCA if he is so inclined. There are plenty of private marshal arts studio that he can take up lessons... also swim lessons as well. There is Spartacus Camp that might go up to his age group. Google them for more info. (It's a daycamp program.)
If I can think of more... I'll post it. Good luck. You might want to post a bit more details of what your son might be interested in so that you can get a more detailed response.