The Edge Interdiction Clique Problem (EICP) aims to remove at most $k$ edges from a graph so as to minimize the size of the largest clique in the remaining graph. This problem captures a fundamental question in graph manipulation: which edges are structurally critical for preserving large cliques? Such a problem is also motivated by practical applications including protein function maintenance and image matching. The EICP is computationally challenging and belongs to a complexity class beyond NP. Existing approaches rely on general mixed-integer bilevel programming solvers or reformulate the problem into a single-level mixed integer linear program. However, they are still not scalable when the graph size and interdiction budget $k$ grow. To overcome this, we investigate new mixed integer linear formulations, which recast the problem into a sequence of parameterized Edge Blocker Clique Problems (EBCP). This perspective decomposes the original problem into simpler subproblems and enables tighter modeling of clique-related inequalities. Furthermore, we propose a two-stage exact algorithm, \textsc{RLCM}, which first applies problem-specific reduction techniques to shrink the graph and then solves the reduced problem using a tailored branch-and-cut framework. Extensive computational experiments on maximum clique benchmark graphs, large real-world sparse networks, and random graphs demonstrate that \textsc{RLCM} consistently outperforms existing approaches.