I am a particle physicist, and I have worked directly on this problem. The uncertainty in the hadronic contributions to the vacuum polarization and light-by-light scattering are large enough that the supposed BSM signal is not significant.
That is, you can do nice high-order paper-and-pencil calculations of Feynman diagrams when the particles involved are electrons and muons, but there are important cases where the particles contributing to this effect are composite: hadrons (which are made of quarks). Since you cannot do calculations on hadrons without considering how the hadron is composed of quarks, you can't avoid getting into strongly coupled quantum chromodynamics (QCD). See here for further discussion: Hadronic Light-by-Light.
That means you can't do your calculation on paper, you have to use a supercomputer and something called lattice QCD. Unfortunately, it's easier to crank out a thousand crappy model calculations of BSM that is supposedly showing up than to properly fund studies of the theory uncertainties. As a result, the precision of the theory values are not good enough to establish whether the muon magnetic moment is consistent with the Standard Model or not.
That said, it's still an interesting place to look, and somebody will work out all the uncertainties eventually. In a few years, there might be something to talk about seriously.