I did not want to drill or remove the top hanger from my lantern, but you could. There are also many types and constructions of lanterns to consider.
I simply mounted the solar light to the side with the light diffuser poking through an existing opening in the lantern's door, holding it in place with a big rubber band. If you cannot locate a suitable color rubberband, try cutting a band from an old innertube.
Because the solar light is still a rather pinpoint spot, I slid a diffuser into the lantern. This one is thin white plastic sheet, but you could also rut a gallon milk carton or similar translucent plastic.