I'll suggest the coat to him, but it may be a hazard in the forest, especially when swimming. Lady could get snagged on a branch under water and possibly drown.
My concern too and that's why I never did put a safety vest on pup for hunting season. Afraid it might snag on something. Lots of people use the neoprene though. They fit very closely. My field trainer does on his Labs but then maybe a bigger, stronger dog is not at much at risk.
This is an article on exactly this topic, keeping them warm with neoprene.