Even I supplied brand and mpn (which I checked via GET inventory_item), I am still getting the errors "The item specific Brand is missing." and "The item specific MPN is missing." when trying to publish the offer.
Any tips how to solve that?
The unexpected solution of this problem was to add a product aspect in your own Language (exactly how the attribute is called in your localized error message) with the same data again.
This let me publish the offer finally.