Welcome to the forums. Did your trailer come with an inverter? If not, no 120 volt outlets are powered when not plugged in to shore power. Probably your best bet would be to find a single bulb light and replace it with a small LED. Use that as a night light, as they don't burn a lot of power. One would burn much less than a tenth the power of a single bulb, cost less than an inverter, and draw MUCH less power than an inverter.
Probably the single fixture with two bulbs in the bathroom would be a good candidate. You'd need two LEDs, but you could swing the bathroom door mostly closed so you can sleep, and the youngster can see there are no monsters.
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