I'd recommend that the tenant be ready to walk away from this rental opportunity. If they feel it is such a great deal that they can't pass it up then they should be even more wary.
Tell the rental manager/owner that you prefer to pay with personal check and that you are perfectly willing to not move in till the check clears. Explain that you will be paying by personal check throughout the tenancy and you want the owner/manager to feel comfortable with your financial situation. Understand that this probably comes from being burned before and while is a bit of an overreaction it can be understood. If they insist then walk away. Fact is you probably don't want to rent from someone so unreasonable. Future negotiation is likely to go the same way over and over. No on accepts living in a win lose situation all the time if they are the loser in the negotiation.