Near-field radiative heat transfer can exceed the blackbody limit by orders of magnitude due to evanescent electromagnetic coupling across subwavelength gaps. While such enhancement has been widely studied in planar and nanostructured systems, the role of engineered meta-atoms in controlling near-field heat transfer remains largely unexplored. We found that the local arrangement of meta-atom pairs modifies the electromagnetic response and opens additional photon-tunneling channels, leading to a significant enhancement of thermal transport. In particular, near-field activated dark modes play a dominant role at small separations, while their rapid suppression at larger gaps results in a strong distance-dependent reduction in heat transfer. As a result, the total radiative thermal conductance can be enhanced by up to 7 orders of magnitude compared to a planar reference, with an additional enhancement factor of up to 32 achieved by varying the intra-pair configuration of the meta-atoms at a separation of 2 µm. These findings establish the light-matter interaction as a powerful degree of freedom for controlling near-field thermal radiation.