According to you, 1-2 weeks in advance of moving house you notified DVLA with a request for a new V5C on 4 March 2023 because yours was 'lost, stolen, damaged or destroyed'. Pl explain.
You did not receive a response.
You did not follow up with DVLA as required.
6 months later a PCN was issued to you at the 'old' address for a road traffic contravention.
Objectively, IMO the fault is yours. You had the legal duty and more than adequate time to get back to DVLA when your V5C was not received. But there's no evidence you did.
So, IMO even if you got the council and TEC to agree on the what, the why would still represent a considerable hurdle.
As said by others, have you contacted bailiffonline?