Might be worth trying Gower Holiday Village at Scurlage - they aren't advertising it but when we were there a couple of years ago in one of their bungalows we noticed a small camping area on the site near the swimming pool. The pool is a bit dated but decent enough, there is an outdoor play area plus an indoor soft play area.
(looking on google earth there is definitely a camping area but possibly caravans only as it appears to be hard standing pitches.)
Amazing family weekend with old steam engines, classic car displays, market stalls, and full catering and bar. And camping on site - Save £25 by booking in advance.