The problem is that you're in Haidian, and police stations there are much more hard@ss than the ones in Chaoyang. This accounts for the variation in what you're hearing--in Chaoyang, I've never had to show more than a copy of the lease with photocopy of landlord's ID attached, but that was an easy-going subdistrict. The landlord's ID is actually a very typical request, but many stations don't require either the original or the landlord him/herself, just a copy of the ID.

I would try to get by with the following:
--your passport (or whoever's name the lease is in)
--relative's passport
--photocopy of lease
--photocopy of landlord's ID (either borrow if landlord is agreeable, or have landlord make several copies, and you stash the extras for future use)
--letter from wuye (management of apartment) stating that landlord does own the apartment and you are a registered tenant. Make sure the exact address that matches the lease itself is on the letter. Most wuye's do this as a regular matter.

Hopefully that will be it, and there will be no mention of fapiaos, landlord showing up in person, etc. Go in with the attitude of confident honored professor at Beida (Tsinghua?) and not a grungy waiguoren student that the Haidian police typically see.