White glue (like Elmer's for example) diluted with water and then painted on the cleaned/dried/whitened bone soaks in and strengthens the skull. I've used it for many years for this purpose; it never yellows or causes problems of any kind. It also leaves the bone looking very smooth and natural, without that horrible artificial appearance that paint or varnish produces.
Varnish or varathane (I like semi-gloss) preserves the colour of the antlers. If you allow them to get into the boiling water or the peroxide, you will lose the natural colour, but a little experimentation to find the right combination of wood stains brings that back nicely. Once they're dry, the varathane preserves them indefinitely.
Bleach will weaken the bone; peroxide is a much safer and more effective method of whitening the skull.