Actually, "a few birds" WILL be missed. That's because in North America in most cases the migratory songbirds are here during their nesting period. And many migratory birds only breed a few times in their lifetimes. So killing them here would have a big effect on species-level replacement because it would disrupt the breeding cycle in an outsized way.
And bees number in the 10's of thousands, per hive. The density of migratory bird is orders of magnitude less, except in some species during the migration, which is almost always a short-period event and not going to result in regular daily bee-predation, so it could be safely ignored.
Of course as I noted above in the case of queens on their mating flight, they are exposed during a once in a lifetime foray, so she is vulnerable. But you are very unlikely to see a queen getting eaten by a bird, so you'd be making a big assumption that the cause of her non-return was birds, vs. say the windshield of a car, or a bug-eating lizard, or your bug-phobic neighbor's weekly visit from the Exterminator.
I'm not trying to argue you out of killing birds - though I can't see why you would choose to do that - just trying to make sure no one is unaware that virtually all song birds and raptors (hawks, owls, eagles, etc., though it boggles the mind that they might be considered bee-predators!) in the US are protected by Federal law. Game birds are subject to both Federal and State hunting laws. Heck, this extends to possession of wild bird feathers, nest and eggs. So, even the relatively harmless picking up of shed feathers, rejected and abandonned eggs, or an empty songbird nest seen in a bush in the depths of winter would get you in hot water without a Fish and Wildlife permit.
The lack of knowledge of this is demonstrated that some people above were openly alluding to violating a Federal Law. Most people really don't know that most birds (and bird parts, like feathers, eggs and nests) are protected. On a home-decoration blog that I visit there was consternation when I pointed out that decorating with wild song bird feathers and nests and birds' eggs (collected and "blown" to evacuate the contents - and such pretty colors!) that was being described, and much-admired, was flat-out illegal. But, it's just "a few" and there are lots more birds, I was told. Nope, still illegal, and more importantly, environmentally wrong-headed.
Surely, beekeepers being in general world-class inventers can figure out how to protect the bees (if not their queens) around their hives from birds? At least enough to reduce the peril to such a low level it was almost negligible, on the order of bee-squashing when re-stacking the boxes during inspection.
Enj.
