We call most of them skylights here, too. "Solatube" is a brand name for the round tubular kind of skylight. As near as I can figure, skylights are square or rectangular, and solar tubes are round.
I have radiant ceiling heat in my house too, and I love it. However, we were able to install a 14" Solatube in the main bathroom several years ago. It does not transfer heat or cold at all. Originally, the bathroom's only source of heat was an infrared lamp; there are no wires running through the ceiling, so installation was no problem. I LOVE that Solatube. Not only does it light up the bathroom, it lights the hallway too. The laundry room is at the end of the hallway, and it has a window. So I replaced the hollow-core laundry room door with an interior glass-paned window, and put sheer lace curtains on it to block the view of the laundry but let light in.
Between that Solatube and the glass-paned door, what was once a long, dark hallway is now light and bright.
p.s. Sandy (the OP) said "If you live in an area with lots of sunshine". You do not need lots of sunshine for a solar tube to work well. I live in the Portland metro area, very gray and rainy, and the Solatube works well.