You can make a cover out of 1 inch foam insulation that will fit the skylight and cover it with tin foil. Use velcro to hold it in place in the summer and take it out in the winter for the shower.
There is a solar tint that will help a lot with the solar radiation heating the inside of the trailer. At the YMCA where I work, we tried it (I thought that it was something that came out of the south end of a bull facing north) and it works. There are a lot of different brands, so look at the ratings.
Also, if your trailer is parked so that the awning is on the west side, in the afternoons, put it down where it shades a lot of the side of the trailer. This will help to some extent.