I could be wrong but as far as I’m aware bus lane fines work the same as fines for going through speed cameras, the authority will send a NIP to the registered vehicle owner who will then inform the authority of the name of the driver so the NIP can be sent to them.
An agency would not receive a NIP unless it’s their own vehicle involved, they may have been informed by their client that they have to pay the fine and now they’re trying to pass the fine onto you, ask them to send you a copy of the official notice so you can check the vehicle details and date of the offence against your records.
If they can’t or won’t let you see the official documentation of the fine tell them to do one, alternatively just tell them to do one anyway.