Free Traffic Legal Advice

I'm a bit lost as to why we're talking about PoFA, in the opening post the OP sets out that they revealed themselves to be the driver in the original appeal, so that ship would have appeared to have sailed.

Do we have a copy of the original appeal - any defence based on disability may be helped by its contents.

As above, Re. counterclaims. Our advice generally centres around helping people make these matters go away as quickly and inexpensively as possible. A counterclaim increases the chances of a hearing being required, and a £3k+ counterclaim would make it a no-brainier for the parking company to send a lawyer. A potential counterpoint is that it might encourage them to make (or agree to, if you propose it) a 'drop hands' offer, whereby they agree to drop their case if you agree to drop yours. It's a risk, and depends on the perceived strength of your counterclaim.

If you wish to pursue a counterclaim, you may wish to seek legal advice from someone with relevant expertise in disability discrimination cases.

If you counterclaim then you will go to court.
If you do not counterclaim, there is a good chance that the case will be discontinued by the claimant before having to pay the court fee. However if you register a counterclaim they can't easily discontinue, or won't do so because your counterclaim requiring them to attend court will still be "live".
Parking companies are not required to follow PoFA 2012 and can issue a claim to anyone they believe to be liable, Believing this to be unfair and having your feelings hurt won't stand up without more substance, so you need to think carefully about your proposed course of action. I can not personally offer any advice on the disability discrimination route you want to follow.

@Sander333 said "appeal as the driver" and "appeal as the registered keeper" which might be confusing. The appeal should definitely be sent as the registered keeper only without identifying the driver, and do not say anything in your appeal in the first person when referring to the driver.

Read https://www.ftla.uk/private-parking-tickets/read-this-first-private-parking-charges-forum-guide/ if this is still confusing or unclear.

File a defence or Acknowledgment of Service by 22 September
If you file an AoS then you have to file a defence by 6 October
You already have the basis of a defence, so don’t lose track of the key dates and see what other suggest as well as posting a draft defence here for comments.

Please read http://www.ftla.uk/private-parking-tickets/read-this-first-private-parking-charges-forum-guide/, in particular so that you can show us the important documents you are talking about.

You are probably aware that blue badges have no standing on private land, unless the land owner defines parking terms which include them.