I'm afraid that you would need to allow too much ammonia to escape (huge noxious corrosive cloud) - just to effect the cooling you are looking for - to make this a proposition of interest.
True!
Also, without any cryogenic distillation of the ammonia I believe it would be unfit for a birch reduction.
Compress the anhydrous NH3 in to a cylinder in a dryice/acetone freezing mixture in a container that the acetone will not dissolve in a deep freezer.