Optimizing a real-life RIS-parametrized wireless channel with a physics-consistent multiport-network model necessitates prior remote estimation of the mutual coupling (MC) between RIS elements. The number of MC parameters grows quadratically with the number of RIS elements, posing scalability challenges. Because of inevitable ambiguities, independently estimated segments of the MC matrix cannot be easily stitched together. Here, by carefully handling the ambiguities, we achieve a separation of the full estimation problem into three sequentially treated sets of smaller problems. We partition the RIS elements into groups. First, we estimate the MC for one group as well as the characteristics of the available loads. Second, we separately estimate the MC for each of the remaining groups, in each case with partial overlap with an already characterized group. Third, we separately estimate the MC between each distinct pair of groups. Full parallelization is feasible within the second and third sets of problems, and the third set of problems can furthermore benefit from efficient initialization. We experimentally validate our algorithm for a 4x4 MIMO channel parametrized by a 100-element RIS inside a rich-scattering environment. Our experimentally calibrated 5867-parameter multiport-network model achieves an accuracy of 40.5 dB, whereas benchmark models with limited or no MC awareness only reach 17.0 dB and 13.8 dB, respectively. Based on the experimentally calibrated models, we optimize the RIS for five wireless performance indicators. Experimental measurements with the optimized RIS configurations demonstrate only moderate benefits of MC awareness in RIS optimization in terms of the achieved performance. However, we observe that limited or no MC awareness markedly erodes the reliability of model-based predictions of the expected performance.