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My NG boiler(s) are in a convection hydronic system - large diameter pipes which go up to radiators and the water moves due to temperature difference (convection). No pumps. My old boiler had a thermocouple in the pilot light which generated enough power to open the gas valve when the thermostat called for heat. My new furnace (recommended because it's more efficient, according to the gas company) has some electronic controls which require 120VAC. When I lost power for 8 days due to Sandy, I got an inverter to drive the 120 Volt electronics from a 12 volt battery. It worked fine.

First, the OEM module output voltage is 1.5V, so you would want one alkaline battery, or a bank of them in parallel, not 3 in series achieving 4.5V as in the unit shown in your third image. Second, LEDs need current limiting, and we don't know if that is happening in the OEM module, or at the individual bulbs, or maybe in the wiring. Some experimentation with a battery and some resistors might help determine whether a series resistor is needed. Start with a high resistance and work down to similar brightness as the OEM module produces. High in this case is probably tens of ohms. You may have to sacrifice a light string to home in on the ideal value. Keep in mind the power requirement for the resistor. P = I^2R.

I found some rechargeable 18650 batteries online ( https://www.batterypkcell.com/18650-battery/ ) that seem like a good fit for the B-Hyve XR. They hold a decent charge and shouldn't be too tricky to replace. Anyone have experience using these with their B-Hyve? It would be great to hear if they keep the system running smoothly through a power outage.