Quitting Physics PhD to become engineer?
There are actually a few areas where you might stand out more than someone with a straight up engineering background.


Look at specific types of organizations like Space research and exploration organizations. These typically have other PhDs in Physics in their program and you might have a better shot. A collection of such roles is here:
http://tapwage.com/channel/space-doctor
Another area are interesting startups. Something like 3D Printing focused startups are looking for bright scientists. Given your experience with mechanics, electronics and tools like Matlab, that could be an interesting area and startups might be more amenable to looking beyond very structured educational experiences as they look for good talent
http://tapwage.com/channel/engineer-in-3-dimensions


Ping me via the tapwage.com site if you have further questions or clarifications and I'd be happy to answer.