If you mailed a check you have to be a bit careful -- the check may show up and may be deposited. If you pay your rent a second time, you need to consider whether you want to stop payment on the first check, or if you want to risk the possibility that the check will arrive at the property management company's offices and be deposited. If you have a healthy bank balance, paying another month's rent early may not be a big issue for you, but many people don't have an extra month's rent sitting in their account. If you stop payment on the first check you will want to be very clear with the property management company that you have done so, and provide them with specific information that will help them avoid depositing the check if it arrives at their offices (including the check number), lest they deposit it and then assess against you a bad check fee.
If your landlord won't waive the late fee, there's only so much that you can do. You can't prove that you mailed a check or that it arrived at their offices. Sometimes a landlord will agree to waive the fee; sometimes a tenant will decide that the path of least resistance is to pay the fee.

