Don't underestimate #2 lead shot.
Might be hard to find in 20 gauge. Don't know.
No one I know of makes 00 buck in 20 gauge.
I think it's #2 or #3 buck that is factory loaded.
Truthfully at s.g. ranges buck isn't necessary to kill an animal under 40 pounds wieght.
Years ago my uncle hutning gray and red fox where he lived in Pa.
Back then, (way back then) the State paid a bounty of 5 bucks per fox killed.
40 plus years ago 5 bucks was near a days wages.
His favorite s.g. was the Remington M-11, a copy of the Browning A-5, in 12 bore with a poly choke, and of course 2 3/4" chambers.
His load was Winchester "baby magnums". A 2 3/4" hi brass hull with 1.5 ounces of lead #2 shot. This in the days before plated shot, buffered loads, shot collars and shot cups.
He took fox out to a full 60 yards and as close and 10 yards with that gun and load.
Even at extreme range the #2 shot would almost always shoot thru and thru a fox.
If you can find #2 shot and perhaps limit shots to 40 yards and less I feel you could take yotes with that load.
As always you want to shoot on paper to check pattern density at the furthest range you will shoot.