In order to beat Carlito, there are two ways I have discovered. Either go into the battle with two pistols, one of which is given to you by Brad and another you can find from a cop, or go into the battle with the sledgehammer or katana. Method one involves running through the Welcome sign and running toward the ice cream parlour, then turn 180 degrees and stand behind the boxes before you go into the seating section. There you can shoot Carlito but he can't hit you. Eventually he will run on top of the food places and you must turn and hide behind the boxes, strafing and shooting him.

Method 2 is to follow the beginning portion of one, except when you get to the boxes, run past them and climb ontop of the boxes that allow you to jump up to where Carlito is. Equip the sledgehammer or katana, then jump up when carlito is shooting at Brad or when he's reloading. Then just smash him, you'll take damage, but his flurry takes down one bar of health, so you should beat him and kill him with one bar of health left.