What you are running into is the fallacy of the 3" 20 gauge. Just because you can fit a 12 gauge payload into the 3" shell doesn't make a 20 gauge into a 12 gauge. I'm betting that those 3" shell beat the h#ll out of you too!
As the bore size gets smaller, the center pattern density gets less and the effective range of the pattern gets shorter. There are many reasons for this. Shot deformation, at least with lead shot, is probably the largest culprit, but increasing shot size as the bore decreases also has an effect on pattern density and the ease of connecting solidly with your target.
The easiest way to carry a 20 and have the performance of a 12 is to find a lightweight 12. The Benelli Ultra Light or an Ithaca Ultra-lightweight are two that come to mind.
The only true way to get 12 gauge performance is to shoot a 10 gauge or 12 gauge gun, limit your range and/or the shot presentations at which you will shoot at birds at longer ranges with the smaller gauge guns.
