I know one needs E&M, classical, quantum, statmech at the graduate level. But what more does one need? Are quantum field theory, relativistic QM, advanced math needed? or just an advanced course in elementary particles on top of basics suffice? Please bear in mind that the intended profession is an experimenter and not a theorist.
I presume this is a study track towards a PhD in elementary particle experimental. To a large extent it will depend on the projects available in the university that offer thesis subjects for graduate students.
If the subject is pure experiment, detector development and/or program development then a strong particle physics course would suffice, because it would be a background for the particular thesis subject . It might be necessary to take a solid state or optics or ...advanced course depending on the detector needs.
If the thesis subject is analysis of data, yes, QFT and advanced math are necessary in order to understand the phenomenological models needed to fit the data.
But again I add that it is something that the thesis advisor should advise on.
To be become a good experimental physicist you must have to have a very good understanding of what is already known. So I would advice you to master the formalism of QFT and the standard model thoroughly. Of-course you would also have to study the experimental technique required.