My enormous sympathies OP. Mine (STEM, RG uni) understood the way the wind was blowing when in first year and applied for literally hundreds of internships. Succeeded in getting internships at end of 2nd and 3rd year. Both actually led to grad job offers. But the horrible process of applying, and understanding what these grad jobs actually means in reality has led to an enormous burnout and her 'year out' to take stock has been followed by not going back to uni to finish the degree. She can't see the point, and is utterly demoralised and exhausted. She continues to get leads via her previous network for start-up roles (don't need a degree). I would LOVE for her to have the piece of paper but she sees this as the sunk cost fallacy and is hard to argue with.
I don't have a single friend whose grad children have been able to find grad jobs. Including an Oxford engineering graduate. (Except the one, who rapidly jacked it in as it was terrible, to go and work for a start-up). A couple have been told to take their degrees OFF their CVs to be considered for jobs.
People do not realise just how hideous it is in the grad job market until their kids are at that point. It isn't about qualifications anymore, but about hard-won experience (which is as hard to get as a job - DDs internships were each a minimum of 5 rounds), ability to network and (probably) who you know. If you have money to help them get set up or take a low paid role in the city with opportunities, that helps too.
My DH was an engineer and he always maintained that without a masters, he wouldn't have been too employable, so possibly an MEng might be helpful if he really wants to continue engineering. If not, then perhaps looking like my DD for start-up type jobs (not very secure, not always well paid, but at least experience, and sometimes pretty interesting) mostly via networking the hell out of LinkedIn?
I don't know what the future holds for my DD, but I sympathise with the difficulty of having an increasingly hopeless adult at home. I understand your prickliness and get it. Good luck.