If a customer is paying you late because they think you won't/can't do anything about it, it seems the same disregard could apply to their payment of late fees.
It’s a lot of effort to keep going so it may just be more hassle than it’s worth. On the other hand, nothing wrong with that process if you’re within the terms of the contract. When the number gets large enough it will get escalated and, when you show your work, get paid and the payment clerk will get their hand slapped.
It's absurd, that's why I like working with smaller companies.