Lead is no longer used in paint, so AFAIK the only realistic-if-tiny issue from ingestion of dry paint is those paints that contain mercury compounds as a mildewcide. And even with those, frankly I flat-out do not believe that chickens would ingest a meaningful amount, unless you do something radically stupid that I am hard-put to even imagine what it could be
Basically chickens do not peck at sound wood and sound paint.
And the solvent part of paint safety is a non-issue, since you would not be putting the chickens in there until/unless the paint is well dried and no solvent aroma remains. Right?
So honestly I see no reason to worry about it.
Most sensible thing is to prepare the surface well (have the wood dry and well-dusted-off, then a thin coat of appropriate primer) and then find some remnants of exterior semigloss in a not-too-dark color to put on. Two thin coats of that, and there ya go.
You CAN get all spendy on the paint if you WANT to, far be it from me to restrain folks from stimulating the economy <g>, but I see no earthly point in it in this case
Good luck, have fun,
Pat