It sounds like the resolution to your problem would be found within the contract. When a landlord requires reimbursement for actual costs incurred on a tenant's behalf there is generally an invoicing covenant contained in the contract. That covenant, whether or not it is reasonable, often governs the obligations of both parties.
I would recommend that you contact an attorney and see if someone will look at your contract and provide you with a brief consultation for a flat fee. If the landlord is in the wrong then you may have damages. If the landlord is right then you will probably have to stop using a bill-pay service and drop off the checks before a late fee would be incurred. Fighting over the reasonableness of a provision in a contract is extraordinarily expensive and therefore few in your situation would ever seek to pursue that remedy.
Next time you want to sign a lease you may want to hire an attorney to review it and make sure that the provisions are reasonable and that the landlord has appropriate reporting obligations so that you have can have the convenience of utilizing your bank to pay the rent.
This foregoing answer is provided by Douglas J Shumway, Esq.: (801) 478-8080. The answer provided is not intended to form an attorney client relationship and should not be construed as legal advice. Mr. Shumway is licensed only in Utah. Mr. Shumway's law firm has offices in Salt Lake City, San Antonio and Las Vegas.