The dog can die, and if you're good with a hammer or axe, you can whack it a few times. It runs off, howls, then returns. When so, hit it some more and it will die.

There are still some options. You can practice on how to hit things with your axe. Basically, that's holding down the left mouse key, move the mouse wheel to the right, then back left again (or move it left, then right). Once you're good at it, take a deep breath and then go after that dog. You can if you want, start off with a beef jerky, making sure he doesn't see you, have him distracted, then whack him a good one, and keep hitting him as he gets back up. He will run off, howl, then come back at you. Be sure that when he runs off, you are in good health. If not, use some painkillers if you have any.
If killing just isn't the thing, try to place a beef jerky on the ground, again, without him seeing you, and as he eats it, sneak by and get the TNT and such.
Remember, you don't have to kill the dog. Sneaking around and staying out of sight works. The object is to progress, not so much to kill.
So, like I said, if killing really isn't the way, the beef jerky, and/or simply sneak around him.