Theory: Since Quote is a battle robot designed to destroy the Demon Crown, it's very likely that he was programmed not to be evil, but simply seeking self-defense, the cost of which probably did not matter to him.
He stole the gun so that he would have some sort of self-defense. Adding to that, if Quote returned to the First Cave, it's most likely that he wanted to return the Polar Star, because, by that point, he had other weapons to use.
Also, judging by the dialog, he trusted anyone who wasn't a threat (Jenka) and tried to help them (by bringing the Puppies, in that case). That's why he only asked for the key and didn't just steal it.