Excuse me, OP. Did not read carefully enough. So you do have a rental agreement, which you said only includes the owner's ID. Normally you also need a signed copy of the 'blue book', which is one of the items required for registering the property in question (where you are living) in the TM 30 system. So your rental agreement is halfway there. No way to get a copy of the blue book before tomorrow? Still, if you are not able to get it, give it a shot tomorrow at immigration, see what happens.
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