The Constructor--Blocker game is played on the edge set of $K_n$. Two players alternately claim previously unclaimed edges. Constructor aims to maximize the number of copies of a target graph $H$ in her graph while keeping it $F$-free throughout the game, whereas Blocker aims to minimize this number. When both players play optimally and Constructor moves first, the final number of copies of $H$ in Constructor's graph is called the score of the game and is denoted by $g(n,H,F)$. Recently, Balogh, Chen, and English systematically studied this game for non-bipartite forbidden graphs $F$. However, bipartite forbidden graphs present additional difficulties. In this paper, we focus on the case in which the forbidden graph is an even cycle. First, using a finite-field-geometric construction, we confirm a conjecture of Balogh, Chen and English by proving \[ g(n,K_3,C_4)=Θ(n^{3/2}). \] We also establish two-sided bounds for $g(n,K_3,C_{2k})$ that relate the game score to classical extremal numbers. Our proofs reveal a structural distinction between the cases $k=2$ and $k\ge3$: a vertex-duplication method works for longer even cycles, but it necessarily creates $4$-cycles and therefore cannot be used in the $C_4$-free game. Finally, we determine the exact order of $g(n,C_t,C_{2k})$ for all $k\ge2$ and all $t\ge4$.