It depends, in Spain there’s a thing called Seguro de Alquiler, (Rental insurance? Not sure the specific name of this service in the US). If the landlord offers a place for rent and has a rental insurance involved, giving them false documents could get you in trouble because of contracts and stuff. But, if the rental is done through a Real State Agency and the realtor (or directly through the lanlord) just asked for it to see but no mention of it in the contract then I think you could get away with it