Ed, Ed, Ed! It's basic ballistics. The only difference in any Ga. shot guns's "lethality" is pattern density at distance. a #6 out of a 12 hits no harder than a #6 out of a .410. If the pattern density is sufficient then any Ga. shotgun is sufficiently lethal for game. If the OP keeps his shots within 40 yds. with his 20 Ga. shooting #5 or #6 shot his chance of needlessly wounding birds is no better or worse than his ability to hit them.
Chief