We consider the problem of simulating a two-sender multiple access channel (MAC) for fixed product inputs, where each sender transmits a message to the decoder over a rate-limited noiseless link based on its input and unlimited randomness shared with the decoder. As our main contribution, we characterize the one-shot communication cost region via almost-matching inner and outer bounds phrased in terms of the smooth max-information of the channel. The achievability relies on a rejection-sampling algorithm to simulate a quantization channel between each sender and decoder, and producing the final output based on the output of these intermediate channels. The converse follows via information-spectrum based arguments relating operational quantities to information measures. Our one-shot results recover the single-letter asymptotic rate region for MAC simulation with fixed, independent and identically distributed product inputs, that was obtained in [Kurri et al., IEEE Transactions on Information Theory 68, 7575 (2022)]. Finally, we extend our result to quantum-to-classical channels with a separable decomposition [Atif et al., IEEE Transactions on Information Theory 68, 1085 (2022)], for which we obtain a similar characterization.