Snake robots enable mobility through extreme terrains and confined environments in terrestrial and space applications. However, robust perception and localization for snake robots remain an open challenge due to the proximity of the sensor payload to the ground coupled with a limited field of view. To address this issue, we propose Blind-motion with Intermittently Scheduled Scans (BLISS) which combines proprioception-only mobility with intermittent scans to be resilient against both localization failures and collision risks. BLISS is formulated as an integrated task and motion planning (TAMP) problem that leads to a chance-constrained hybrid partially observable Markov decision process (CC-HPOMDP), known to be computationally intractable due to the curse of history. Our novelty lies in reformulating CC-HPOMDP as a tractable, convex mixed integer linear program. This allows us to solve BLISS-TAMP significantly faster and jointly derive optimal task-motion plans. Simulations and hardware experiments on the EELS snake robot show our method achieves over an order of magnitude computational improvement compared to state-of-the-art POMDP planners and$>50 \%$better navigation time optimality versus classical two-stage planners.