The paint you're planning on using is already rated for exterior use. I can't speak to its quality but if used as per the manufacturer's guidelines you should get their expected service life which is often in the region of five to seven years, long enough I'd think for a bird feeder.
Most painting guides suggest that to extend service life in exterior applications it's vital to prime the wood first. This may no longer be as true as it once was because of modern paint formulation, but it is generally sound advice and usually you won't regret priming before painting.
If you want to maximise the time these will look their best there's something else you can do before the paint goes on, and that is to seal all end-grain surfaces to prevent moisture ingress. Epoxy is ideal for this purpose. You can use standard epoxy glue, no need for anything specialised (and more expensive) than this. Rub the thoroughly mixed epoxy on to all end-grain surfaces and once fully hardened lightly sand and then paint as normal.
but is considered environmentally friendly/safe for birds
All finishes you might use are bird-friendly once cured.
Almost all finishes these days aren't inherently toxic (no lead pigments and so forth) and a cured film of the binder is quite inert by definition.
So there's no need to go with any green or 'green' paints for this. Standard paints of various types are used for beehives and insects are far more sensitive to chemicals than even birds are.
