"May I carry the bag?"
I think "Shall I carry the bag" would work for the ましょう ending too. I don't know if duo accepts it though
May I carry the bag?
Shall I carry the bag?
The first asks persmission, the 2nd makes an offer of doing something (which uses the same ending as saying that you should do something collectively, so context determines which makes most sense).
That's not... really... what it translates to, unless you give us an explicit context. You're never asking for permission here, you're just asking whether or not you'll carry the bag, and even 'you' isn't really all that strongly implied there.
Something like 鞄をお持ちしてもよろしいですか？ might be the translation...