The only logical reason I can see that they would not accept a business check from a charitable organization is that they are concerned about check fraud. In such a case, I fear that even a cashier's check would be unacceptable to them since cashier's check fraud has become a greater concern in the banking industry lately. They are likely unfamiliar with the Church (i.e. they don't trust it) or have gotten burned too many times with third party checks regardless of the source.
Another option would be for a member (under the direction of the bishop, of course) to purchase the money order themselves, pay the rent, get a receipt, and then submit that to the ward clerk for reimbursement. We use a procedure like that occasionally for utility bills or gasoline as well.