Functional analysis and real analysis are not (entirely) pure math, they are basic and highly applicable in application including but not limited to theoretical physics. The more important and harder to answer question is how a limed budget of time should be allocated to learning an unlimited amount of useful mathematics. My entirely unhelpful opinion is that between 3 and 30 percent of mathematics learning time should be devoted to functional analysis and real analysis depending on individual needs, abilities, and interest.
I'm not in high energy, exactly, but I say learn what you are interested in. I don't know what level you are at. I think quantum mechanics provides nice motivation for functional analysis, so it's good to learn them together, or quantum mechanics first. Maybe functional first, if you are more mathematical, but I kind of look down on that, even though I am a mathematician (some PDE might also suffice in place of QM).
If your style is very mathematical, you can be like John Baez and know limitless amounts of math, but work in quantum gravity (although he quit that). If your style is more towards physics, I suspect you can probably get by if you almost ignore math, except whatever you need to learn along the way. So, it's really whatever you are interested in, I think.