Bee eaters can put a big hole in your bee population, especially when they start to bring all their friends along. I've seen them near hives and they have not worried the bees, maybe because there was alternative tucker around. I believe they catch flying bees, so a net won't help. Hopefully you can discourage the ones you have before 40 or so friends and family join them. I'm not sure if gas guns work on them but that's another suggestion to try. I doubt cats will get up in the trees to chase the bee eater.
Let us know how you go.