I don't really see a delivery job working for someone like Brun who lives by her set routines. Doing Pizza deliveries means navigating to endless new addresses, and constant interactions with new strangers in new environments. Brun was so thrown by her sudden new habitations that she forgot something simple like taking a shower. I think making deliveries would be hellish for her, never being able to inhabit her comfort zone.
Theoretically, Brun's job at the Whale will eventually open back up. Meantime, she'll want another situation involving set routines, I'm thinking. Not sure what it would be, though. Now, if she were to become a US Mail carrier (as opposed to delivering pizzas), there's lots of routine in that; always going back to the same addresses, and less need to deal with humans. But those kinds of jobs tend to be hard to get, I think.
WRT integrating Brun into the main cast, I think the path is fairly clear. She'll encounter Clinton again (maybe at COD), and there'll be a conflict between Renee and Clinton (probably played for laughs). Perhaps Brun will eventually became a COD regular.