Indicating that you are under age 59½ (and not dead or disabled) means that code J applies to a regular distribution no matter how long you have had a Roth IRA. A financial institution has no business asking if you have met the 5-year waiting period. That's for them to determine based on their own records.
I only suggested the prospect of changing custodians/financial institutions because the one you have now seems somewhat incompetent (or at least uncooperative) based on this experience you are having with them.
(Regarding the selections for the answer to the question asking which describes your situation, they don't entirely make sense. It's extremely unusual for a Roth IRA custodian to use anything other than code J for a regular distribution to the participant who is under age 59½, relying on the participant to make the exception claim on the participant's tax return rather than the effectively certifying though the use of code T any assertion by the participant other than the participant's age.)
