Modified will be far too tight for most shooting...
I shot Hevi-shot for most of a season, back when I could get it cheap (I purchased a bunch at about 50ยข per shell).
At first I could not hit a darned thing, and opening up a shell and seeing that slag, I thought it was probably patterning poorly... Took it out and patterned it, and it was just the opposite. It was patterning far too tight, and opening up to an improved-cylinder or even a skeet choke, my success improved greatly.
At least in my gun, and IC with Hevi-shot at 40+ yards was plenty tight.
Interesting that you cannot get it to pattern in your turkey gun though? I am using 1 3/8 oz loads, and it patterns plenty well out to 40 yards (as far as a good turkey pattern) with a standard full choke.
How far are you lookin' to shoot, and what chokes are you running it through?