I think that's where the independent advice/solicitor comes in for the OP.
Also, get everything in writing/electronic copies of any objections and rebuttals from them. As in I'd e-mail your concerns and BCC your own Gmail/whatever address. If you have a meeting where they try something on as in the above post, claim that technically they can bill for it (I mean we don't know all the details) then again send an e-mail to confirm that they've told you they want to do X etc.
If you've done the legal minimum then part of this is a business decision for them re: billing the customer but the falsifying results part is something you don't want to be involved with and I'd put that in writing in an e-mail and keep a copy, refuse that part of it. Getting a customer to pay a high price for some work is one thing, defrauding them/falsifying stuff is another ball game entirely - you might reluctantly want to do something unethical/objectionable to save your job there (and keep a record of your objections) but don't cross the line into doing anything illegal and do seek independent legal advice on this to try and ensure you don't inadvertently break the law either.
Ultimately it's not a nice situation to be in, you want to protect your own job but you also need to stand up for yourself and push back on this stuff, you've gotta do it constructively, try and offer solutions that don't involve any illegality but ultimately be confident in saying no to certain things and pushing back hard on obvious red lines. Perhaps some fudge could involve you offering to be proactive, call the client, talk through their concerns, sell the work you have done confidently but also get sign off from your current employer for a day or two to spend at the client site to finish it all in order to reassure the client and make sure you're doing the job properly. Currently, they're down £12k of revenue from this client surely they can afford a day or two of your time to recover that properly.
Internally you need to push this as a future business thing, the client isn't stupid, you want to retain them and repair relations with them for the sake of not just revenue they're refusing to pay but future revenue - you management is being incredibly short-sighted if they're going to try to openly rip them off - not only are they showing themselves up to be cowboys and damaging future relations but also opening up the company to legal action, the client just carrying on refusing to pay etc..