We report experimental progress toward demonstrating background-limited arrays of membrane-isolated transition-edge sensors (TESs) for the Background Limited Infrared/Sub-mm Spectrograph (BLISS). BLISS is a space-borne instrument with grating spectrometers for wavelengths lambda=35-435 mu m and with R=lambda/Delta lambda similar to 500. The goals for BLISS TESs are: noise equivalent power (NEP)=5x10(-20) W/Hz(1/2) and response time iota<30ms. We expect background-limited performance from bilayers TESs with T-C=65mK and G=15fW/K. However, such TESs cannot be operated at 50mK unless stray power on the devices, or dark power P-D, is less than 200aW. We describe criteria for measuring P-D that requires accurate knowledge of T-C. Ultimately, we fabricated superconducting thermistors from Ir (T-C >= 135mK) and Mo/Cu proximitized bilayers, where T-C is the thermistor transition temperature. We measured the Ir TES arrays in our 45mK base temperature adiabatic demagnetization refrigerator test system, which can measure up to eight 1x32 arrays simultaneously using a time-division multiplexer, as well as our single-pixel test system which can measure down to 15mK. In our previous Ir array measurements our best reported performance was NEP=2.5x10(-19) W/Hz(1/2) and iota similar to 5ms for straight-beam TESs. In fact, we expected NEP similar to 1.5x10(-19)W/Hz(1/2) for meander beam TESs, but did not achieve this previously due to 1/f noise. Here, we detail improvements toward measuring the expected NEP and demonstrate NEP=(1.3 +/- 0.2)x10(-19)W/Hz(1/2) in our single-pixel test system and NEP=(1.6 +/- 0.3)x10(-19)W/Hz(1/2) in our array test system.