Instead of using the mask, you could duplicate the object the planet is orbiting around on a different layer. Put one layer above and one layer below the planet in the stacking order and then use opacity to switch between them.
This can be done in Inkscape relatively easily, and the following method is similar to how the Shape Builder tool works in Adobe Illustrator.
Make sure everything is ungrouped, and there is no clipping mask just a regular circle instead to be used to cut everything out.
Make a combined path of the drawing plus the circle, by using Path > Combine Ctrl/...
Your type shapes are compound shapes, i.e. shapes with "holes" or "counters".
Use the Direct Selection Tool (White arrow) to select only the inner paths and delete them. Then make the mask. Or simply delete the inner shapes from your second image.
You have to click the actual path for the inner shapes. For the "O" you'll need to delete 2 paths as show in ...
In the Transparency panel... you have clicked the art thumbnail so it's active, right?
You can tell the active thumbnail by the colored border around it in the Transparency Panel.
If the mask thumbnail is the active thumbnail (meaning you are working on the mask) then ONLY the mask will move, regardless of the chain icon.
Basically, if the mask is active ...