Here is the response I recieved from Support:

The custom report that you would like to build at the moment will not be possible unless a joined report is created and the only issue is that we will not be able to add the contacts to the campaign from the joined report or be able to export it. In regards to setting up a custom report type, this will not be possible due to the relationships that the three objects you are looking to include share. The way that custom report types work is that the B object is related to the A object and if you add a C object, it will be related to the B object. We unfortunately cannot have a B and C object both related to the A object in the custom report type.

The issue here, as we saw, is with the object relationships of accounts, contacts and this custom objects district products. The way to pull this report would be if the district product is related to both the account and contact. The way that I was able to find that you would be able to do this would be through custom formula look up fields. My recommendation would be to create in the Product District custom object a formula look up for the account relationship field and the product relationship field. I believe it would be possible to have the district product automatically associate to both the account, contact and product through a cross object look up formula field. With the same information appearing within the account and the contact, you can then run the Contacts with District products report to obtain the date you are looking for.