Enough batteries will power anything, it's just a question of charging them back up in the alotted span of time, particularly if you go solar.
As a starting point, you might look at the input voltage on your device. I suspect you use some sort of AC-DC transformer to power it. If you could power it directly with DC, that would probably be a bit more efficient.
Then you'd need to figure out how many amp-hours you'd need to power the device while you're sleeping.
Then you'd decide if you want to take enough batteries for the whole trip, or charge them.
Then you'd decide on which type of batteries to take and your charging method.
It's nice that you have sites that prohibit generators. I've found that sites without electricity tend to be more annoying because half a dozen people have Harbor Freight generators running all day.