The nature of the degree isn't always that important, however, maybe in engineering it is.
Physics is a critical part of engineering, but then so are many of the other sciences. So the biggest stumbling block you'd have to get across is, to be blunt, detailing how your physics degree is of more use than a specific engineering degree. I think that might be a challenge.
As has been mentioned, engineering is a huge and varied field. You may have elimnated civil engineering, but then there's mechanical, electrical, computer, naval, acoustical, corrosion, aerospace, automotive, computer, electronic, petroleum, systems, audio, software, architectural, biosystems, biomedical, geological, industrial, materials, and nuclear.
Each is a very specific field with their own courses and as stated above, physics is one part of that, but engineering in any field is a practical application of that and many other sciences as well as economics and other fields.
The key is practical application, employers will want to see you've applied the knowledge to a set of circumstances and solved it. If you've looked at physics beyond theory, that may help and you could emphasise the experimental or applied physics aspects of your degree.
But to take something like ESA, where your degree may be attractive, the astrophysicists would be working more on observation from, say, a telescope, interpreting what is seen and using physics to confirm theories, add to theories, or learn more about the universe.
The engineers would have designed and built the telescope.
Completely different fields.
I don't want to put you off, but you are competing against peopel who will have specific engineering based qualifications and experience. In mostcases, for example chemical engineering, an employer will not entertain anyone who hasn't read Chemical Engineering. In addition, if you're applying for all engineering jobs, it can show a lack of understanding to those you're applying to as not all engineering is the same.
Just make sure you research each job you apply for and what that field requires, then find what aspects of your degree could be useful there. If you have practical design and build experience, that's great and use that in all applications.