You are correct about resizing the cone and back boring they too can impact the pattern and reduce recoil. ballistic specialties does a great job and will send you new chokes and back bore as well. some loads pattern different than others.
My father an engineer atIBM reloaded shells he also shot in the skeet leage. he enjoyed greatsuccess and shot 500 straight in all gauges several years while competing.
He had specific powder, wad. primer.shot combinations for all gauges to patern the way he wanted for skeet, trap, and hunting
I want give them all away but here is what works great in 12
AA Shot Shell, Federal Primer,AA wadsAA 540 powder 215/16 drams (not made anymore) 1 1/8 #9s in a skeet barrel. (skeet, first season doves, quail)
Peters Blue Majic shell, CCI primer, AA red wads, 3 3/4 drams Red Dot powder, 1 1/4 of #6 worked well as aduck load in the 34" full trap #8 for long range trap shooting.

try different loads with modern heavy shot a traditional lead see what the difference is. but most important see where the gun shoots