Lynn,
What about a "face" sunscreen (not necessarily a kid's one) that is sweatproof and waterproof? I have used the Neutrogena one and the blue Coppertone stick & bottle with no problems.
Part of the issue may be that you're putting on and he's expecting a problem? I "switched strategies" with my kids. Bought a "new" bottle and told them it was "special" for faces (and eyes) - which it is. And then we put it on AT HOME while we brush our teeth. It dries better because they aren't already wet or sweaty, and I don't have to hold them down or risk smearing it into their eyes while they are anxious to go play. This strategy has worked perfectly! It's cool in the house. They don't expect it to be a problem. And we've been fine. If/when we need to reapply when we are out and about, I just use the stick...I ONLY reapply after they are seated, have/are having a snack and have dried off and cooled down a bit. We used this "break time" to potty, eat and reapply before going back out for some fun.
BTW - I keep the "face" bottle in my bathroom next to my deodorant. It helps remind ME to put on sunblock before we all go outside.
I hope this helps!
Sara
P.S. Can you tell I also work at a pool?