If any of the bills (preferably utility bills) come in your name, you can use that to help establish residence, though having your name added to the lease is the best course of action. The landlord should allow the tenant's lease to be amended provided your presence doesn't violate the terms, though keep in mind that if something goes wrong you'll most likely be on the hook financially.
But is this really necessary? You're looking at a lot of paperwork and fees (driver's license, FID, etc ) for what is potentially a non-permanent residence should the two of you decide to get a bigger apartment or buy a house. I'd leave it alone for the time being, but YMMV.