Why does this night light cause the GFI to trip?
A GFCI device compares the current flowing on the "hot" to the current returning on the "neutral".  In principle, the only way a GFCI can trip is if current finds some third path.  

What about battery charging? Isn't the current going into the battery and not returning?  NO.  Think of a battery charger like an irrigation water pump driven by a water-wheel.  All the current going into the water-wheel falls out the bottom.  That turns a shaft.  And on the other end of that shaft, the pump pushes a completely different stream of current.  The GFCI is akin to having current meters on top and bottom of the water wheel proper, to make sure all the water that enters the wheel, exits.  It has nothing to do with the pump current. 

So, if your nightlight has only 2 connections, hot and neutral, then there is no third path - every electron which enters via hot must exit via neutral.  A GFCI trip should be impossible.   That would seem to leave 2 possibilities. 


Something about the electronics of the night light is creating noise or signal delay in a way which is irritating the GFCI.  This could be exacerbated by a defective or poor-quality GFCI. 
There actually is a ground fault, but it's somewhere else and the problem is being triggered by the presence of the nightlight.  Picture a broken receptacle where the neutral is very close to the ground, and plugging the nightlight in shorts neutral to ground.