How do I find out what is under this ceiling cover for hanging a Charleston/Joshua Swing?
This isn't a complete answer, but neither is it appropriate as a comment ("to ask for more information or suggest improvements")... Just a few suggestions as to how you might get a little more information about the structure.

There's probably some eave overhanging the wall outside the space. The height of the fascia would probably reveal the vertical dimension of the rafters, because typically the eave is simply the result of the rafter tails extending beyond the wall.

Edit: The above may or may not be a useful approach. When I wrote it I was thinking of a soffit detail in which the rafter is simply cut vertically, as in the first drawing below. Not surprisingly, it turns out that there are other ways of doing this -- if the rafters are tall then the tails might also be trimmed horizontally to reduce the fascia height and move the soffits upward as in the second drawing.




Another strategy is to drill a small hole in the ceiling somewhere. Probe into the hole with a wire to measure the depth of the space inside and come to some conclusion about the thickness of the rafters.

If a visual of the space is important consider renting or buying a borescope, also known as an endoscope. These can be had as cheaply as USD$20 for a USB model which you'd view with a computer or mobile device, or at a higher price for one with a built-in screen, higher resolution, etc.

You'll need to know something about the spacing of the rafters as well. This also might be apparent at the exterior eaves. If you can see subtle undulations in the roofing surface these would also reveal the spacing between the rafters. Visible fasteners holding the bead board to the ceiling may also, and invisible fasteners might be detectable with a magnet.

When you know more about the structure you'll be in a better position to decide whether to "wing it" and just hang the swing, or whether some engineering might be appropriate.