The key here is that the ‘may I take your baggage’ implies that you are a service person. You are correct, てもいい is MUCH more common for casually asking if something may be done. But you will never hear anyone in pretty much any job in Japan say てもいい to a customer.
To break this down easily, think of ましょう and おう (the casual version) as statements, rather than literally meaning let’s. Literally all they do is turn a sentence into a statment. Think of it like putting emphasis on the I’ll, in I’ll grab a drink.
I’LL grab a drink … as for you? (the as for you bit is implied)
I’LL get you bags… will I? (the polite form just making it clear it is a question)
This is why this grammar point also means ‘shall’, because that is really all it ever is. The question form is only created by your tone of voice.