Dodge...
No TSP. As you found out it's too harsh.
Quick run down: we clean the bones as much as possible, sometimes lightly simmering them. The we use either Sal-Soda (actually a "mordant" used in dying fabrics) or we use Hydrogen-Peroxide (H2-O2), but we get that at a beauty supply store where it comes in different "Volumes" (strengths). We use Volume 20, which is strong but controllable. It degreases and whitens the bones. Sometimes multiple treatments are needed.
Oh yes... safety first with bone dust... stinks really well, too!!
If you want to pursue it more my wife would be happy to make a tutorial and send it to you.
The pics show a few things... The little knife I made for the wife. It has bone scales and black fiber spacers and brass pins and some file-work. The bone was found, so it was cleaned by Ma Nature. It polished up like ivory on the edges. The long flat piece is a weaving shuttle. It also feels like ivory. I sliced that on a bandsaw (free-hand) and shaped it (gently) on a small belt sander, then by hand. The bird skull shows how H2O2 can whiten bones.
Sorry for the not-so-great pics. I need a better camera and bone is a really hard thing to get a good picture of... must be the way it reflects light, etc.