From my understanding, absent "ちうわけで", this simply means, "Is it OK if I go to break." Is it similar to "こういうわけで"?
ちう (and ちゅう) is a slurred version of という, so ちうわけで is the same as というわけで.
というわけで basically just means "So, ..." or "For that reason, ...". It's mainly used to describe a conclusion after stating a reason. It's also used to draw the listeners' attention before moving on to the main topic (e.g., "So, let's start today's lesson"). Sometimes it's used even at the very beginning of an exchange (like English "So, hello, my name is ...").
From this answer:
So with that, …
This is especially common in TV or radio shows when the host needs to press the show forward or move on to the next topic. The という wraps up the preceding statements into わけ and essentially means, "Now that all this has been said, let's move on."