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We present Gemini-S and {\it Spitzer}-IRAC optical-through-near-IR observations in the field of the SPT2349-56 proto-cluster at $z=4.3$. We detect optical/IR counterparts for only nine of the 14 submillimetre galaxies (SMGs) previously identified by ALMA in the core of SPT2349-56. In addition, we detect four $z\sim4$ Lyman-break galaxies (LBGs) in the 30 arcsec diameter region surrounding this proto-cluster core. Three of the four LBGs are new systems, while one appears to be a counterpart of one of the nine observed SMGs. We identify a candidate brightest cluster galaxy (BCG) with a stellar mass of $(3.2^{+2.5}_{-1.4})\times10^{11}\,{\rm M}_{\odot}$. The stellar masses of the eight other SMGs place them on, above, and below the main sequence of star formation at $z\approx4.5$. The cumulative stellar mass for the SPT2349-56 core is at least $(11.5\pm2.9)\times10^{11}\,{\rm M}_{\odot}$, a sizeable fraction of the stellar mass in local BCGs, and close to the universal baryon fraction (0.16) relative to the virial mass of the core ($10^{13}\,{\rm M}_{\odot}$). As all 14 of these SMGs are destined to quickly merge, we conclude that the proto-cluster core has already developed a significant stellar mass at this early stage, comparable to $z=1$ BCGs. Importantly, we also find that the SPT2349-56 core structure would be difficult to uncover in optical surveys, with none of the ALMA sources being easily identifiable or constrained through $g,r,$ and $i$ colour-selection in deep optical surveys and only a modest overdensity of LBGs over the extended core structure. SPT2349-56 therefore represents a truly dust-obscured phase of a massive cluster core under formation.