Okay, my mistake.
I haven't gone through Dead Money on an unarmed playthrough yet, but I can see that the limb damage bonus the Bear Trap Fist has would help a lot. When you did that playthrough did you not use any ranged weapons because the Police Pistol has a 0 Guns Skill requirement.
Honestly, I couldn't help but use a ranged weapon. Speakers made damn sure of that. I used one of the randomly spawned Laser Pistols instead of the Police Pistol, though. Given that the character had about ten points in both Guns & Energy Weapons, it made sense. I don't count Throwing Spears because I can't aim them outside of VATS due to my inexperience. Does the Police Pistol have less spread than the Laser Pistol by default?
Typically, I was critting every three out of four hits and killing all ghosts except Trappers on a critical with the Bear Trap Fist. Best part was that I didn't have to aim for center of mass like I do with Power Fists. VATS helped quite a bit, especially once I got Dog/God's perk. Course, a lot of that was Finesse, Better Criticals, Paralyzing Palm, Super Slam, Stonewall, Purifier, & all the other standard Unarmed perks.
-Nukeknockout