I'm not in the account department, but a place I used to work for handled it thus:
Let any unidentified moneys remain unallocated.
Let the overdue notices go out to customers who haven't had payments allocated.
When the customer rings to complain that they have paid, get them to tell you the payment details, identify and allocate the payment.
"Oh, and in future, if you put your account number on the payment we can avoid this confusion. Thank you have a nice day."
I have a category in my personal accounting I call Deadbeat Vendors. For instance, an insurance company stopped sending bills to my home. For some reason, they thought sending the bills to the insured site where there is no mailbox, no office, no staff, and no means of receiving mail would be a great idea.
Meanwhile, I have a control panel that alerts on expected invoices. The insurance bill popped up, the bill didn't come in, and I went to the company's web site to check.
The policy was renewing and it only bills ten months out of the year. I didn't know if I owed or not.
According to the web site, no bill was due. And then I got mail to my correct address saying they had charged a late fee because the account number had changed, something I could have known if they had sent the bill to the address that's worked for years.
Deadbeat Vendors are kept paid a month in advance to make sure they have nothing to whine about, and I send them a letter explaining why they became deadbeats in my system.
It's a win-win. They get paid in advance. I get to heckle.
It's not simple mistakes that earn deadbeat status. I make mistakes, too.
Mistakes made with church lady degrees of sanctimonious piety earn the Deadbeat award. The electric company whose bills often arrive on or after the due date and then tell me to be responsible, the water company who believes it's OK to send postcard bills the post office mangles and doesn't deliver, and this idiot insurance company.
Amazingly, and probably to the credit of deadbeats everywhere, I've gotten replies harumphing about company integrity. But, just like the company can't remove the late fee for the bill I proactively checked on before the due date, my systems won't allow me to remove the deadbeat flag.
Sorry, Buttercup, your company is a documented deadbeat for at least a year. You can appeal then. Have a nice day!