Sounds like, to your knowledge, they are wrong about my being able to challenge?
To explain all the ins and outs would take ages, so I'll cut to the chase: you can arguably reclaim the penalty from the lease company, but you have to be willing to take the lease company to court.
In short, you can only challenge a PCN if you
don't pay it, payment deprives you of the right to appeal. Where they say "
Legislation requires immediate payment by the registered keeper" they're being incompetent and thick: legislation gives the registered keeper 28 days to pay or challenge, they should have challenged by requesting a transfer of liability.
The bottom line is lease and hire companies are incredibly thick and stupid and they won't accept they're wrong unless and until a court tells them, the corollary of this is that unless you're actually willing to take them to court, there's no point in doing anything else. Sending letters threatening to take them to court without following through would be a waste of time.
The question therefore is: are you willing to have a massive row with the lease company and take them to court to enforce your rights?
In parallel, you could contact the council and ask them to refund the payment on the basis that you want to make representations, and on the odd occasion this can work. The problem is that if the council simply ignore you or say no, there is no recourse open to you.
Also as a general rule, involving MPs in PCNs is a waste of time, they're more likely to mess things up than anything else.