The forum posters raise issues for discussion between us to try and flesh out whether there's a defence, what the tactics might be and how the OP might proceed.
The CEO's photos are evidence which would be presented at adjudication and which are available to the authority when considering your* representations. We should see this evidence which sits on their website and all we need to access is the PCN number and VRM.
*- you posted 'Not my car but my blue badge...'.
IMO, before going further, who is the registered keeper? This is essential because a Notice to Owner would be sent to them, not the driver.