Having to pay for transactions beyond 60 days is not common since it depends on factors that come under 205.6 (fraudulent use). What you disclose in your card agreement and whether or not the claim falls under the requirements of 205.6 is something you should determine before going any further with your investigation.
In any event, an unauthorized transaction can occur in other ways than what you described. Does this person have a spouse/child/friend who could have had access to the card, used it, then put it back? If the person used the card without permission, it's unauthorized. Did the cardholder make a transaction at the store that day but not the one that posted to the account (clerk made an error in which card was charged during a busy day)?
The investigation you've done up to this point is a great start, but to base your case solely on that information is very shaky and IMO does not qualify as a reasonable investigation since you have other quickly accessible avenues to obtain more information (sales draft request from the merchant). I'd ask the cardholder the above question about other persons. If she refuses to answer, then you can use her refusal in conjunction with what you have discovered to conclude that you will deny the claim. If she states no one had access, then I'd recommend requesting the sales receipt and reviewing it for validity.
If within 10 days you determine there is no error, the letter must state:
1) You have concluded no error has occurred
2) A description of how you concluded there is no error (This does not have to be a step-by-step explanation. Using this example, you could explain that upon reviewing the account history, the transaction information, the delay in contact, and the cardholder's unwillingness to provide essential details, you have concluded there is no error.
3) No provisional credit will be issued (Some people consider this statement optional. My opinion is that it's one sentence and completely removes any doubt that there will be any credit)
My opinions do not necessarily reflect those of all the voices in my head.