I have a Wallas and I have worried about that as well. I haven't run it all night, but I have sat out storms for a day or two and run it quite a bit. I have two interchangeable motor/house batteries on my boat that I can switch between. I always keep one fully charged battery switched off for one reason only, to start the boat motor and the other being the house battery that I use for the Wallas, fans, radios etc. If this battery runs out of juice, I still have a full battery to start the motor. I also have solar panels to help keep the batteries full.
I have found that when I use my Wallas as a heater I prefer to leave it on low with the lid open and use a Camfro fan, which has very low energy draw, which I have mounted above the Wallas to circulate the air. I prefer this as it moves the air around better, helps to keep the windows clear and is quieter than the Wallas fan. I also have a Ecofan that can be set right on the Wallas burner and uses no electricity to operate the fan.