As of Patch 1.0.3, this is still probably the optimal thing to do, but it's nowhere near as great as it used to be. Feel free to just hold on to the crappy gun and shoot things with it if you want, I guess.
You can sell it to an unsuspecting dupe of a merchant who will overpay you for it! It sells for quite a bit of cash, so long as you sell it immediately. Specifically, Heodan, in the Caravan Encampment in the tutorial is a fairly incompetent merchant. He will buy your things for substantially more than they are worth in many cases. For example, he will give you 75 cp for some clothing that is worth next to nothing. Even better though, he will give you 150 cp for The Disappointer. That's a lot of scratch for just starting out in the Dyrwood.
The trick is that this means you need to retrieve the gun and sell it to Heodan before triggering the ambush at the stream, (i.e. before interacting with the waterskins found near where you'll pick up The Disappointer). This isn't a big deal, but it's a thing to be conscious of.