Hello Tino,
The only way to mark a invoice paid by using a date in the past is using the manual payment option in the AP Module. The Manual Payment option will look at the Due on Date of the invoice to assign the Paid on Date. The manual payment option will force you to use a check number, please DO NOT make the check number higher then the number that prompts, otherwise this will disrupt your check number sequence.
If you have any question on creating/applying manual payments, please let me know and I will have a member of the support team reach out to assist.