If you have a clear non violated back on the osage bow no backing should be needed. If you want to add one for your own comfort TBIII works great for rawhide backings. Be sure the rawhide and the bows back are oil free before adding rawhide to the back. I wash the rawhide with warm water and Dawn dish detergent. Be sure to rinse well with warm water. I scrub the bow back with Dawn and warm water and rinse it with boiling water. Add a thin, even coat full strength glue to the bow back and the rawhide. Also, I like the rawhide to be in 2 pieces putting one on each limb overlapping them at the handle. With both pieces down I smooth the rawhide out squeezing excess glue out along the sides and wipe the excess off with a damp cloth. With TBIII I use strips of old bed sheets to wrap the bow/rawhide working from the handle out the limbs. Be sure the rawhide stays centered as you wrap. I leave a wrap on for about a bow for about an hour. That's usually enough time for the glue to set up be not be completely dry.
At this time I check for air bubbles and excess glue pockets and work them out toward the sides with your thumb. Then use a sharp razor to trim the excess rawhide along the sides.
It good to give the bow a few days at least to dry. The wood has absorbed moisture from the glue/rawhide and even though it may feel dry on the surface you have to give the wood time to release the absorbed moisture. After the bow/backing is completely dry you can add a thin layer of TBIII over the rawhide as a moisture barrier then add whatever finish you plan to use for the bow.
I would tiller the bow first then add the rawhide before adding the finish.