The Lost and Damned is more suited to the Minimal Kill Challenge than The Ballad of Gay Tony. I have attempted both already!
A couple of years ago played the Lost and Damned through to the final mission with a total of 6 kills. The final mission added about 40 kills and I never made a final save or completed all the side missions!
Recently played up to This sh*t's Cursed
and so far unable to complete with zero kills due to some of those pesky Immune-to-non-player types.
Regarding The Ballad of Gay Tony, I've had a nibble at this as well and found it very difficult with a much higher kill count. This is the least trampled path if your looking for some unexplored ground. I didn't venture very far...