According to the Superex website, your inverter has 1200 watt surge protection. They don't say exactly what that means, (e.g. how long can it handle 1200 watts?) but at least they have tried to do something.
Even without any surge protection, you should be ok because an electric blanket is a purely resistive load and will not require the high inrush power of a highly reactive load such as an electric motor.
I think the bigger question has to do with overall energy consumption. Best case, a small inverter will be 90% efficient, so you will be drawing 360/0.9 or 400 watts from the battery bank. At 12.5 volts, that is 32 amps. If it stayed on full time for an 8 hr sleep, that is 256 amp-hours. I assume the blanket has a thermostat and is not on full time, but I suspect the energy use will still be high. How big is your battery bank?
Dave has looked into and posted on this problem in detail. Check his previous posts. If you are careful like Dave, you can make it work, but it is easy to go wrong using 12v batteries for heating.
As far as your furnace fan goes, you should be able to look it up. The standard Atwood small furnaces draw about 2.9 amps. I purchased the model 8012 with a low power fan that draws only 1.8 amps. (And is somewhat quieter) Once you find the power draw, estimate the duty cycle and compute the nights energy use. We tend not to use our furnace at night as it wakes us up. We preheat the van with the furnace, use two down comforters during the night, then in the morning we heat the van again with the furnace.
Charlie