There are two possibilities I'm thinking about. I'll put them in separate posts so people can refer to the individually.
First, towels. LTF has lots and lots of white towels. There is no fee for using LTF towels. If you go to the outdoor pool you are supposed to bring your own towels from home for that, though.
What if the Coopers had snuck some of the towels home, maybe some dry ones so they could protect the car seats from the girls wet swimsuits if they didn't change to dry clothes at the club? Maybe they did this regularly and returned the towels later. If they had some of those towels at home BC could have used them for cleanup or for lining the car trunk. They could have been washed once at home (or not at all) and then brought back in a gym bag and added to the big baskets to get washed again.
Holes: He would have had to check in to bring the towels in. He would have had to carry the bag in (suspicious if he was supposed to be searching for his wife) Also, he had the girls with him, so he would have had to leave them in the car or bring them in with him.
All in all, seems risky for him, but in panic, maybe?