The bottom line is this: The property manager, as the landlords’ agent, can evict you for any delinquency or lease violation, with due process of law. And from the sound of it, even though you may believe you are only delinquent on the late fees and not the monthly rent, it doesn’t make much difference. You’re still in breach, and therefore the landlord has grounds to evict you.
From a legal point of view, there is no differentiating between the rent and the late fee.
If you are delinquent on late fees, you cannot rely on having paid your monthly periodic rent for relief. The landlord can apply your fractional payment to the past due rent, or to the late fees, and it doesn’t matter either way. You are in breach of your lease agreement no matter which way the landlord goes.
Furthermore, from the landlords’ point of view, if you’re having trouble scratching together 100 or 200 dollars in late fees in a month’s time, that’s a worrisome sign that you may be unable to make your rent payment in the future.