It looks like they are playing very dirty but I don't see how I can prove it or what I can do about it
There
is something you can do about it. And, no, they're not playing dirty, just being their usual incompetent selves.
We see cases like yours a lot on this forum, and there is a backstop legal process to get the matter reverted to the Enforcement Notice stage.
After the CC, TfL must register the PCN at the Traffic Enforcement Centre, (the "court" which has no judges and no courtrooms). Only when this is done can they start to collect the debt using bailiffs. But they must give you a final opportunity to pay by sending you an Order for Recovery. This will demand the CC amount plus the TEC fee of £10. Included with this will be the TEC form to submit a Statutory Declaration, (Form PE3). You can then submit this, ticking the box for "
I made representations about the penalty charge to the local authority concerned within 28 days of the service of the Notice to Owner/Enforcement Notice/Penalty Charge Notice, but did not receive a rejection notice."
TEC will then cancel the OfR, and the CC and the matter will revert to the stage when you submitted your reps.
You must now be proactive and after the period for payment of the CC expires, contact TEC about every 7-10 days to find out if the PCN has been registered. Alternatively, you can look at the audit trail for your PCN on the TfL website to see if the PCN has been registered. You can then not bother waiting for their OfR, but download Form PE3 from the TEC website, fill it in, get it witnessed, (that's when you sign and date it iwth witness present), and send it off by email or post.
https://www.gov.uk/government/publications/form-pe3-challenge-an-unpaid-penalty-charge-notice