We investigate the interaction dependence of the liquid-gas critical point of symmetric nuclear matter in finite-temperature lattice effective field theory. Building on the pinhole-trace algorithm, we benchmark a first-order perturbative treatment for representative Hamiltonian splittings and then compute the finite-temperature equation of state for a sequence of sign-friendly lattice Hamiltonians ranging from an SU(4)-symmetric interaction to Hamiltonians with physical [1]S0 and [3]S1 channel dependence and three improved leading-order Hamiltonians. The finite-temperature analysis is complemented by zero-temperature calculations of the symmetric-matter saturation point and the binding energies of selected nuclei within the same lattice framework. We find that the benchmarked perturbative strategy is quantitatively reliable in the thermodynamic regime studied. Across this Hamiltonian sequence, the LO Hamiltonians improve the overall description of finite-nucleus binding energies and move the zero-temperature saturation point toward the empirical region, while lowering the critical temperature from 15.33(6) MeV to 13.50(17) − 13.71(19) MeV. These calculations show that finite-temperature criticality is not fixed by zero-temperature saturation and binding alone, and provide a complementary benchmark for future lattice interaction development.