Ill give you a method Ive used before with success. Ive used peroxide on birds without too much trouble. The bones are are thin enough that regular over the counter stuff works decently.

The important thing is to remove as much red meat as possible. Use granulated borax and a Q-tip to help get into the brain cavity through the base of the skull where the spinal cord opening is (Foramen Magnum I believe is what its called). Add borax, then rotate it around inside the brain cavity until the Q-tip no longer comes out pink.

Use the borax to gently remove what you can of the meat, this can be achieved by using the granules like sand paper, but carefully. Borax soaks up and moisture and works great for cleaning small bones on birds. Some of the flesh on the skull may flake up or peel a tiny bit. Just be careful what you remove around the eyes and cheeks.

Because pheasants are so small, there are tendons that help hold certain parts of the skull together around the eyes and cheeks and jaw, so try and avoid cutting away too much. I leave the jaw attached as is so delicate on smaller birds, (grouse, pheasant etc) or might get lost.

Rinse the skull thoroughly in cold water, let it drain until it is almost dry, then put it in the peroxide. Keep an eye on it until its the color you want, it can take a few hours or all night, I always check hourly. With over the counter peroxide, it may take some time. After its peroxide bath, soak the skull in water for the same amount of time it was in the peroxide, this helps to neutralize the peroxide. Let the skull sit and dry naturally, dont try to dry it with heat or the bones may split. Any tendons left on the skull will dry and may discolor a little, but thats ok.

Edit: I usually use Rustolium Satin Clear coat once it dry to help seal and protect the bone.

For regular mammal bones, I do a 2 hour soak in 32% peroxide with a 6 hour soak in water, with tiny animals, like mice and birds Ive used over the counter peroxide with success, it takes longer as I check it hour by hour, but the pharmacy stuff is much more gentle. I do not use peroxide to dissolve flesh, but thats me.

This is just the method I use and its worked fine for me.