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My clients typically invoice their customers via email with me in "cc". They include their invoice (bearing my stamp), a clear copy of the POD and a copy of the carrier confirmation. If any of the three parties is not satisfied with any of the documentation, we know right away. For example, if the copy of the POD is not clear enough, the customer (broker) says so right away and carrier sends a clearer copy within a matter of minutes. This approach also avoids the issue of invoices not sent out in a timely manner. I think it all works quite well.
Sounds like you've got a good system in place...