I would say talk to graduate school advisers at various schools offering terminal masters programs. Some schools don't put too much emphasis on the formalities of degrees, just your level of comprehension and number of classes taken in physics. If you have a solid enough foundation you may be able to get a masters degree in 2-2.5 yrs, where the first year may be spent "catching" up with graduate level courses. Again this depends on the type of graduate school... my advice would be to not shoot for a top 20 school to get a masters (almost none of them offer terminal masters), because they'll most likely turn you down. Try a good state school.
Since you've already taken some solid mechanics/dynamics courses (which you've hopefully done) for your engineering degree it wouldn't hurt to pick up some more mathematical/abstract books on classical mechanics (i.e., Taylor, Goldstein, Marion & Thornton). Know your Lagrangian and Hamiltonian dynamics. Courses you'll lack in is in upper-level E&M (take a look at two very common undergradute text separately authord by Griffiths and Purcell), Quantum Mechanics I and II (look at Griffiths undergraduate text... Quantum II is also referred to Atomic and Nuclear Physics), and Thermal/Statistical Physics (you may have taken a course in thermal physics, but this course would be more mathematical... text I used was Reif). These courses are more commonly taken during Junior and Senior years for physics majors. They are the standard for any undergraduate. The courses mentioned are not (in my personal opinion) all that rigorous... just tedious. Further, you should be acquainted with vector analysis/calculus (know Stoke's theorems, Gauss's Theorem, Green's Theorem, etc. and you should be good at solving ordinary differential equations because you'll need these basics to learn how to solve partial differential equations). I say these are "basics" because most undergraduates take more advanced courses like E&M II (E&M waves, Special Relativity, radiation), Adv. Classical Mechanics (non-linear dynamics, systems of particles, continuous systems, etc.), Astrophysics (a bunch of stuff about stars that has little physics in it), and General Relativity (generalization of special relativity... way mathematical and more often taught in graduate school). There's way more classes many undergraduates take to satisfy their interests, but for you excelling at the "standard" is a must for any graduate school.
I've personally done some research about doing graduate studies in a field you don't have a degree in, which would be mathematics for me. I am a bit more prepared though... I am 3 courses away from getting an undergraduate degree according to most top 50 schools. I hope to forgo a BS degree in maths for a MA in maths, where I will be also working on a graduate degree in physics (MS or PhD... not sure yet). Yet, my first year of maths would be "catching" up with the other students while I complete the graduate physics standard. The second year would be finally working on graduate level maths. Schools I could do this at? Found some after a bit of research and web browsing, which is what you need to do, too.