The challenge of identifying the best feasible arm within a fixed budget has attracted considerable interest in recent years. However, a notable gap remains in the literature: the exact exponential rate at which the error probability approaches zero has yet to be established, even in the relatively simple setting of $K$-armed bandits with Gaussian noise. In this paper, we address this gap by examining the problem within the context of linear bandits. We introduce a novel algorithm for best feasible arm identification that guarantees an exponential decay in the error probability. Remarkably, the decay rate -- characterized by the exponent -- matches the theoretical lower bound derived using information-theoretic principles. Our approach leverages a posterior sampling framework embedded within a game-based sampling rule involving a min-learner and a max-learner. This strategy shares its foundations with Thompson sampling, but is specifically tailored to optimize the identification process under fixed-budget constraints. Furthermore, we validate the effectiveness of our algorithm through comprehensive empirical evaluations across various problem instances with different levels of complexity. The results corroborate our theoretical findings and demonstrate that our method outperforms several benchmark algorithms in terms of both accuracy and efficiency.