Insulating a post and beam construction roof/cathedral ceiling
The tongue-and-groove wood planks conceal insulation above them, probably fiberglass batts between the rafters. As you've observed, they are clearly insufficient, because the room is too hot. There are two safe ways to improve the situation:


Add more insulation underneath the existing ceiling wood, then install a vapor retarder (retarder, not barrier. A smart membrane, not poly sheeting), then cover that up with drywall.
Remove the roofing materials and install 4-8 inches of rigid foam insulation board above the roof decking, then new roof sheathing over that, then new roofing materials.
Build a metal roof (white or bare metal) over the existing shingle roof, using eave-to-ridge 2x4s to build ventilation channels under the new roofing. Such a roof will reject the vast majority of the sun's rays, which is the primary driver of heat gain.


Since you don't want to mess with the roofing, that leaves option 1. If you don't want to ruin the look of the ceiling with the tongue-and-groove boards, you're out of options and will have to live with the situation.