GLEAM-X J1627-52 was discovered as a periodic ($\sim$18 min) radio signal over a duration of three months in 2018. It is an enigmatic example of a growing population of 'long-period radio transients' consistent with Galactic origins. Their nature is uncertain, and leading models invoke magnetic neutron stars or white dwarfs, potentially in close binary systems, to power them. GLEAM-X J1627-52 resides in the Galactic plane with a comparatively coarse localization ($\simeq$2 arcsec). Here, we study the localization region to search for spectrophotometric signatures of a counterpart using time-domain searches in optical and near-infrared imaging, and MUSE integral field spectroscopy. No sources in the localization display clear white dwarf spectral signatures, although at the expected distance we can only provide modest limits on their presence directly. We rule out the presence of hot subdwarfs in the vicinity. We found no candidate within our search for variability or periodic behaviour in the light curves. Radial velocity curves additionally show only weak evidence of variation, requiring any realistic underlying system to have very low orbital inclination ($i \lesssim 5$ deg). Two Balmer emission line sources are reminiscent of white dwarf pulsar systems, but their characteristics fall within expected M-dwarf chromospheric activity with no signs of being in a close binary. Currently the white dwarf pulsar scenario is not supported, although longer baseline data and data contemporaneous with a radio active epoch are required before stronger statements. Isolated magnetars, or compact binaries remain viable. Our limits highlight the difficulty of these searches in dense environments at the limits of ground-based data.