1) Flat out refuse to agree to late fees. They get removed from the deal or the whole thing is off.
2) Simply pay the original amount and ignore whatever late fees you have added on. Don't like it? "Speak to accounts", "we don't have a process for that", "these fees aren't on the paperwork that I have here", "speak to your point of contact", "I don't know anything about that", repeat.
It all comes down to who needs who more. A large company has a mountain of suppliers to choose from and they are often fairly easy to replace. They have potential suppliers trying to get in the door all day long so if you don't like it you can show yourself out. Meanwhile a small supplier only has a relatively small set of large customers they could possibly do business with and they are very hard to get.
That is laughable. The only thing it gets you is ignored. The most aggressive late fee I've ever seen agreed to in any contract is 1.5%/mo starting on day 121 (original was 2%/mo on day 31). Any company I've worked for that has had more than $100MM or so in revenue has simply refused to sign any contract with any sort of late payment penalty, but they would sign 2/10 net 30 contracts all day long. Some companies even require that their accounting department follow any terms for a discount.
In college I was part of a junior enterprise (basically an outsourcing firm for students), and we figured out really quickly that the due date was the pay date after consistently applying late fee logic. Saved us a lot of back-and-forth.