What is the correct order for a payment breakdown?

Escrow, Interest, Principal, Late Payments?

It would depend on your legal contact with your customer. I am not aware of any legal requirement to apply payments in any specific order.

While Texas does not have a rule or regulation that specifically addresses the application of payments, the late payment charge in Texas is limited to one late charge per payment and no interest in addition to a late charge, and you could easily run afoul of that applying payments first to a late charge. Additionally, it is a practice that if it ever came before a judge or jury would be hard to argue is not abusive and or deceptive.

But, the application of payments have nothing to do with whether or not a consumer makes a payment for the amount due. If a monthly payment for $300 is due and if the consumer makes a $300 payment, what difference does it make how it is applied as long as it is applied according to the contract. Just because you choose to apply payments to fees, interest and then principal, if they make a payment at their scheduled payment amount, they will not be late again. Pyramiding of late charges also violates Federal law.

So it doesn't really make a difference until a less-than-full-amount payment is made. Then it matters a lot. Since most systems apply payments the same way every time, wouldn't it be best to set it up to take late charges last all the time?

Yep.

If "a less-than-full-amount payment is made", they are late again no matter how you apply payments.

If are not properly separating two things here. What is considered a payment due (contractual amount) and how those payments when received are applied. They are two very distinct things.