We propose a finite BSC-decomposition method for representing an arbitrary binary-input symmetric-output channel (BISOC) as a mixture of binary symmetric channels (BSCs) with at most $\ell+1$ distinct crossover probabilities. We show that the weighted average capacity of the component BSCs exceeds the capacity of the original BISOC by at most $(π^2\log e)/(48\ell^2)+O(\ell^{-5/2})$ bits, and the leading term is asymptotically optimal in the worst case. Based on the finite BSC-decomposition, we design a channel simulation scheme for arbitrary BISOCs. Combined with efficient channel simulation algorithms for BSCs, our scheme achieves a communication rate with $O((\log n / n)^{2/3})$ redundancy over the capacity for simulating $n$ copies of a BISOC.

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