This work presents ThermoMesh, a passive thin-film thermoelectric mesh sensor designed to detect and characterize spatio-temporally sparse heat sources through conduction-based thermal imaging. The device integrates thermoelectric junctions with linear or nonlinear interlayer resistive elements to perform simultaneous sensing and in-sensor compression. We focus on the single-event (1-sparse) operation and define four performance metrics: range, efficiency, sensitivity, and accuracy. Numerical modeling shows that a linear resistive interlayer flattens the sensitivity distribution and improves minimum sensitivity by approximately tenfold for a $16\times16$ mesh. Nonlinear temperature-dependent interlayers further enhance minimum sensitivity at scale: a ceramic negative-temperature-coefficient (NTC) layer over 973-1273K yields a $\sim14{,}500\times$ higher minimum sensitivity than the linear design at a $200\times200$ mesh, while a VO$_2$ interlayer modeled across its metal-insulator transition (MIT) over 298-373K yields a $\sim24\times$ improvement. Using synthetic 1-sparse datasets with white boundary-channel noise at a signal-to-noise ratio of 40dB, the VO$_2$ case achieved $98\%$ localization accuracy, a mean absolute temperature error of $0.23$K, and a noise-equivalent temperature (NET) of $0.07$K. For the ceramic-NTC case no localization errors were observed under the tested conditions, with a mean absolute temperature error of $1.83$K and a NET of $1.49$K. These results indicate that ThermoMesh could enable energy-efficient embedded thermal sensing in scenarios where conventional infrared imaging is limited, such as molten-droplet detection or hot-spot monitoring in harsh environments.