Authorship Dispute: Soft threats from funders to cancel project unless they are listed as coauthors
Moo You need to get a new job, and you need to submit this work for publication the second that these people no longer have any power over you. It sounds like they have copies of all of your things, which unfortunately means that if you just leave and don’t do anything you could wind up with them publishing your paper(s).

If they don’t have the ability to publish your work, you have more flexibility. I have no idea what your professional situation is like or if your work is time-sensitive, but you do have the serious option of delaying publication until you’re further away from them. There are pros and cons to each option, it’s pretty context dependent, but if you can take the hit for not publishing, letting the paper languish they might drop it. On the other hand racing and publishing it without them might make them give up. It depends on the psychology of the handlers, but if you can guess the lengths they’re willing to go, you might able to get out of the situation without having an authorial dispute. Both choices could also lead to further conflict: either ten hounding you about it indefinitely when you don’t publish the paper or them contacting the journal and trying to formally dispute the paper.

Your main goal for an authorship dispute should be to get as many walls between you and the handlers as you can. Straight up don’t talk to them as much as you can. They’re already in violation of their own policies and of the EU University policies (it sounds like there’s a partnership between the non-profit and the EU University), which is a great start. Contact the ethics office at the EU University and explain the situation, including providing any documentation that you have of what work they did or did not do. From then on, you can forward any angry emails to the person in the ethics office working with you. If the non-profit has an ethics office, then you can use them for this purpose too. You should also talk to the ethics office at your university and feel out how receptive they would be to helping you out. This kind of filibustering is incredibly effective in all kinds of disputes.

However, the best wall you can put between you and these people is the journal itself. Submit your paper(s) to journals that have clearly defined authorship definitions, or even better require submissions to document authorial contributions. At the end of the day the journal is the final arbiter of who gets on the paper and who doesn’t.

Another option would be to fake conceding. Split the work up into multiple papers and divided it such that the stuff they’re claiming they touched is all in one paper and the core concepts are all in the other. It’s easy enough for the joint paper to languish in revisions while your other paper gets published. How effective this is can wildly change depending on the exact circumstances though.