I wouldn't trust that beast if I was you. That thing is a killer.”— Phil
There are two ways to complete this quest:
Recruit the dog as a companionEdit
There is only one known way of completing the quest without killing the dog, and that is by giving it some iguana-on-a-stick. Some can be found in the locked fridge that the Crypts guard. Once given, the dog will follow you and help you in battle. Phil will warn you about the dog saying that it is a killer, and you will gain 550 XP.
Kill the dogEdit
There are many ways to kill the dog in the demo, but the most quick and efficient way would to get close to it, and open fire with a burst from the H&K MP9 10mm SMG. Phil will be grateful that the dog is gone, and you will receive 335 XP, a bit less than completing it the other way.