Here's my opinion - I think the Embassy will consider your application separate from that of your cousin's. Therefore, another original copy of the tenancy agreement has to be submitted for his application if you can't wait for your visa to be released together with the original supporting docs. You also have to consider that there's a huge number applications to be processed and you can't expect the Embassy to look for your file just to view the original tenancy agreement when they work on your cousin's application. This may actually cause some delays....
