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This nonsense is another reason I always opt for paper statements. There should be a law requiring any entity that offers on-line billing to make available 5+ years of past statements for both current and past account holders.
As of now, statement retention and access policies are all over the map. Heck, I've run into multiple instances of companies updating their computer system and dumping all the old on-line statements in the process, despite originally promising X months worth being accessible.
A more common issue, beyond the typical 12-24 months retention, is that often when one's account is closed, access to all statements is promptly lost. Similar goes for many financial firms too, including banks and on-line trading sites.
Shame to read Comenity won't assist you. You could try requesting 3, wait a bit, and then request another 3. Might work. A more drastic route would be litigation against Comenity and/or some party relating to your use of that account in order to obtain a subpoena for the statements; contacting an attorney for assistance.
As for the lates, if they were spread out over time, that may be viewed more favorably than a string of lates, though it depends. Sending a goodwill letter to Comenity, repeatedly if need be, asking for the lates to be rememoved may work, but could take many months before success, if ever all.
Another route is disputing the lates with the credit bureaus - may not eliminate the lates (especially, if they're legit), but could be another way to get access to the past statements.
This exact situation is the reason I've created folders in my Google Drive account for every card and bank account. Every month, I save a pdf copy of the statement for each of these accounts to their respective folders.
I never used to save these types of documents, however, the one thing I've learned since I decided to take my credit and finances seriously is to save what you can. You never know when you need to go back to something in the past.