Exact conditional tests for contingency tables require sampling from fibers with fixed margins. Classical Markov basis MCMC is general but often impractical: computing full Markov bases that connect all fibers of a given constraint matrix can be infeasible and the resulting chains may converge slowly, especially in sparse settings or in presence of structural zeros. We introduce a SAT-based alternative that encodes fibers as Boolean circuits which allows modern SAT samplers to generate tables randomly. We analyze the sampling bias that SAT samplers may introduce, provide diagnostics, and propose practical mitigation. We propose hybrid MCMC schemes that combine SAT proposals with local moves to ensure correct stationary distributions which do not necessarily require connectivity via local moves which is particularly beneficial in presence of structural zeros. Across benchmarks, including small and involved tables with many structural zeros where pure Markov-basis methods underperform, our methods deliver reliable conditional p-values and often outperform samplers that rely on precomputed Markov bases.