>>536063591
Just use any weapon with low aiming time. Assault rifle, then charge rifle are good.
Charge rifle has 2s CD, 1.25s aiming after penalty, so firing cycle for normal pawn is 3 seconds, and for careful shooter - 3.25. You only lose about 8% DPS (or I'm bad at math), but shooting becomes more stable. Plus a little time for 3 shots at 300 RPM - so the real DPS loss will be even less than 8%.

If the pawn is good for something other than combat, try to find a unique weapon with a bonus to fast aiming - you can reduce the penalty even more. Fast aim weapons are useless on shooting specialists with 1% aiming time, but here they shine.

Even if you want to use careful shooter as a specialist, you only need three specialists to cycle the buff.
The role itself gives -50% aiming time, that is 75% on our careful shooter. The buff from another specialist gives another -50%, so that's 25%. Maybe not ideal, but not bad.