Must a renter pay late fees for late rent payments if the lease contains late payment penalties?

On our lease agreement Page 1 paragraph 1 it states ' rent must be postmarked by the 1st of the month. If the rent is not received in a timely manner and the rent is not postmarked by the 1st a late fee of $20 per day will apply. Page 1 paragraph 2 states 'Monthly payments may be paid by check until the first check is dishonored. Time is of the essence and no excuses will be accepted. Rent shall be made payable to landlord and hand delivered (or sent by mail at thants risk" to landlord at (address provided) Any rents lost in the mail will be treated as unpaid until received by the landlord. Page 1 paragraph 5 continued onto page 2 states 'Tenenat hereby acknowledges that late payment will cause landlord to incur costs not contemplated by the rental agreement, the exact amount which will be extremely difficult to ascertain. In event the rent is not postmarked on or before the 1st of each month, regardless of cause including dishonored checks, tenant further agrees to pay a late charge to landlord equal to twenty dollors ($20) each day rent is late. Neither ill health, loss of job, financial emergency or other excuse will be accepted for late payment.
I am pretty sure that covers us, this renter did not have a check post dated until Jan 3rd, they were notified via email January 5th that payment had not been received, they called immediately and said it was in the mail. On the 8th we received a letter from tenant containing a bill of ours which was post marked January 6th (sat) and received Jan 8th (mon) on Monday the 8th, we had still not received the check, the tenant was informed they were to drop off a payment immediately including a $100 late fee, which we decreased and should not have. They received the said rent mail today Jan 12 (fri) the tenant has called stating there mail was marked undeliverable, the same address they have was used on the 6th and received in 2 days including a Sunday. I need help how to deny this tenants request for the late fee to be returned with out having to have this get ugly as they have threatened in a voicemail. ALong with previous threats to tell future renters what horrible people we are. ANy and all help would be appreciated, as they are military and we would prefer not to have to get the husbands superior officers involved to solve this matter.

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