and received an email to confirm
This hinges on the date and time at which TfL acknowledged that your renewal application was successful. As I understand it, this does not work retrospectively, in other words of your old BB expired on, say, 10 Dec, and your new badge became effective on 11th but TfL did not put your vehicle on their exempt lost until 14th then you were not exempt between 11th and 14th.
These dates are critical, pl put them all up front. We need to know whether you have a defence based in substantive grounds or only mitigation.