You should be fine with a photocopy of your passport just in case police is checking.
If the police officer is not satisfied (very unlikely) you may be asked to go with the officer to the place where the passport is safekept to show it. If that place is considered too far away the police will certainly find a way to verify your identity - might be a hasssle then but not a serious problem for you !!!

(e.g.: if your accommodation is in a different city they may send a local officer to that place and check the passport - you would certainly permit the hotel staff to hand over the passport to a police officer, donīt you ??? )