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We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the "LSM anomaly" possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. We thus obtain universal bounds on the local operator content of (1+1)$d$ conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a $\mathbb Z_N\times \mathbb Z_N$ symmetry realized projectively at each site. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain non-trivial bounds on charged operators when $N$ is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between (1+1)$d$ symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.