There are several good ways to kill Metzger, but you ain't got the necessary equipment. The first thing to do is lock all the doors . . .
Here are a few tips:
(1) Go to the Den. Get a SMG; I prefer to do so by having a female character named Buffy and getting lots of presents from Rebecca, but you can do it any way you like. If you don't want a female character, or one named Buffy, Tubby has a SMG about half the time. Kill the sonofabitch and take it, or trade for it. Then walk up to Metzger (behind a locked door), and fire one burst. That should do it.
(2) Get a hunting rifle (also available in the Den). Attack either from Becky's back room, or from the windows of the Slaver's Guild. Using some APs to hide from return fire should allow you to avoid most return shots. This is also possible with the shotgun, though it's more difficult.
However you do it, Psycho maxxes out your damage resistance and gives you an extra action point, so you should use it if you can. The easiest way to kill him, however, is to walk right up to him and burst-fire the sonovabitch. Behind locked doors, as has already been discussed.