The problem that I faced in this quest was:
Sinjin had a gun to Kent's head and my approach to the scene was from the front. I tried everything I could think of to take down Sinjin with quick strikes with melee weapon, a really powerful sniper rifle but nothing was working. My dialog had already caused Sinjin's henchmen to flee. So all I had to do was take down one guy.
So I needed time to get more shots with a very high damage. And for me Jet did the trick. The 10 second slowdown of time helped there. Secondly I needed to boost my damage, because none of my rifles or pistols were working. Psycho provides a 25% boost to damage.
So with that in mind this is how I finally completed the optional objective.
- Have a really good high damage, low firing rate like pistol or rifle.
- Have some Jet, Psycho or any time slowing/damage modifying chem.
- After the third intimidation prompt as the Silver Shroud jump into your inventory and pop the Jet + Psycho. I popped one Jet and 3 Psychos. Got really close and went into VATS and lined up shots to the head.
Make sure you get close to the platform, so that you get a high VATS rate for the head. VATS modifying/AP boosting/critical hit gear is a bonus for this.