I do not know how you could not find a program that accepts people from a math background. I just picked a university at random: e.g. see North Carolina State University, Physics Graduate admissions. NCSU clearly states they have considered applicants with electrical engineering and mathematics degrees as well as physics. NCSU goes on to write that they have also considered applicants in the social sciences as well. I suspect other programs also examine qualified applicants in mathematics as well. The problem you may have is to convince an admissions committee that your specific mathematics courses suggest you will be successful in their program.
The rest of your post is also puzzling. For example, you mention getting a physics related job and later transitioning to experimental work in physics. In practice, employers are very careful in their hiring. They realize they are making a lifetime committment. In my experience, it is easier to get into a graduate school than getting a job. The committment is less < 10 years, for the graduate school.
The good news is you may be able to get a "physics related" job without a full graduate degree in physics or at least an allied graduate degree in place of physics. For example, Ohio State University, offers a program in geodetic engineering/geodesy. Geodesy would allow you to use a (very) strong background in boundary value problems (potential theory), and linear algebra (adjustment computation and least squares), and mathematical physics in a rewarding career. You need not know circuit theory or antenna theory or quantum mechanics in graduate physics.
But it is unclear in your post, would you be happy outside of experimental work. If you have limited experience in experimental work, how do you know this is what you want to transition into. Are you a hobbyist. Do you do electronics, arduino, chemistry experiments. Did you like chemistry labs, physics labs, building telescopes, etc. Most experimental physicists I know were "hands-on" types.
The OSU program may even have some gravity measurement/ operational component, that may satisfy your "experimental" criteria desire. Geodesy is not just theory, it has some practical component. Job outlooks in this area are quite good. It is a lot less competitive than the physics academia route, although it is also professionally challenging.
My suggestion is to reword your post and give us a little at a time. Are you trying to get into a program or a job. Are you using your aptitude to learn physics on your own backing up your interest in pursuing a career in physics, in which case your attention to your desire to enter the experimental/laboratory aspects, of this career is a non-sequitur.
