Here are the details on bookings like yours:
https://www.airbnb.com/help/article/245/pay-for-a-longterm-reservation
All bookings 28 days or more are subject to the above.
So, that means you have to pay for a full month up front, and the remainder a month or so after you check in.
If I were you, I would try to change the booking to 2 bookings: One for 27 days and the other for 4 days. For bookings less than 28 days, you should be given the option of splitting the payment into 2 installments. Not sure about the other 4 days; it might depend on the dollar amount whether it can be split or not.
How long ago did you make this reservation, and what is the cancellation policy? You may not be able to cancel without losing money if the cancellation policy is Strict. You can try to cancel online and it should tell you if there will be a cancellation fee before you actually cancel.
If it looks like there will be a cancellation fee, try calling CS back and have them do the 2 reservations. You can point them to the above link for the payment details on 28-or-more day bookings. Sometimes, CS is learning at the same time as the rest of us.
I agree with you that what you are asking is perfectly reasonable, and that Airbnb should be helping you to do this. I think it just has to be set up in a specific (if unintuitive) way.