Extended domain-wall networks that emerge in moiré materials provide a distinct platform for quasi-one-dimensional electronic states. However, the interaction-driven orders in confined networks remain largely unexplored. Here, we discuss superconducting (SC) correlations in interacting helical domain-wall-ring networks in the closed topological domains formed within the moiré patterns of an underlying twisted bilayer honeycomb lattice. We first analyze the system within the framework of an infinite-size theory and show that inter-ring SC-pair tunneling is renormalization-group relevant in an appropriate interaction regime, favoring enhanced SC correlations through inter-ring phase locking. To address finite-size effects resulting from the ring-network geometry, we present a self-consistent variational approach. Our analysis shows that even in the regime where the infinite-size theory predicts strongly coupled pair tunneling, the induced phase-locking scale remains strongly suppressed. In contrast, the SC scaling dimension is governed mainly by density-density interactions and ring geometry, decreasing modestly with decreasing twist angle while remaining insensitive to the pair-tunneling strength. This discrepancy demonstrates that ring networks cannot generally be understood as simple finite-size truncations of their infinite-size counterparts but can exhibit qualitatively distinct collective behavior. Our study highlights how the interplay of confinement effects and ring-network geometry can reshape SC correlations.