First, use TWO 6V golf cart batteries wired in series for 12V. Placed them on a 3-4" thick piece of foam to keep them off the ice.
Second, get a 12VDC fan.
Third, spend the money and get a pure sine wave inverter. 1000W should be more than enough for what you want to run (no fan) unless I am guessing wrong on the satellite box.
Fourth, I am not sure if LCD TV would do well if you left them there and the temperature drop to -20F.
If the batteries are fully charged (make sue you have a GOOD muti-stage charge) when you leave and you put them back on the charge for a couple of hours when you return. I would not worry about the batteries. You do need to check and fill them with distilled water. Of course, hauling them back into a warm cabin would be better.
