Jackie,
Your room looks beautiful by the way, and it actually somewhat resembles my great room in my house which we built specifically for southern exposure and western protection. Like yours, my great room is a double height or two story ceiling that both helps to visually connect the upstairs and downstairs, but also helps circulate natural convection throughout the house.
I guess the part that would concern me would be those skylights. My great room has lots of windows, especially on the south side, but some on the east side as well. In my case, this means that in winter, the low sun pours through the southern windows all day and in summer, as the sun rises much higher, those southern windows are essentially shaded, limiting the accumulated heat.
But in your case, the skylights are going to defeat any potential shading. They look beautiful and let in lots of natural light, a feature I built into my house, but understandably problematic with summer heating. I would think that a potential simple solution would be to put some type of shade under the skylights. Would it be possible to attach a thin sheet of white fabric to the internal frame of those skylights? My thought is that a thin fabric, almost like cheesecloth, would reflect a lot of heat right back out of the skylight, but would still be translucent and let some indirect light through. Potentially you could still get lots of natural sunlight (and personally I love natural sunlight), but not so much heat and I would think that the floor would not heat up nearly so much.
Just as a thought, I really do like the idea of passive solar homes as they are beautiful, comfortable, inviting and make good use of solar
energy. However, I doubt that the heat stored in the summer will last till winter. Rather I think that the heat stored in the day will last into the night, but this is but a minor point.
Other than shading the underside, I guess you could tint the skylight, but then that defeats the purpose of the skylight in the first place. I strategically planted
trees on the east side of my house to shade the eastern windows in the heat of summer, but the south side is in the clear and I don't know how you could plant trees to block the sun at noon in the summer what with the sun being more of less directly overhead. I guess you could simply replace the skylights and cover with traditional roof, but this is both expensive and cuts into the architectural charm of your house. But my best guess is to prevent direct sunlight from striking that (beautiful) floor in the first place.
Your room has such a similar layout to mine that I almost want to attach a picture for comparison, but for the moment I don't have one handy.
I wish I had better ideas, but the shade fabric is the idea that stands out to me the most.
Good luck and I hope this helps. Please let me know what you think.
Eric