IMHO, as someone who has always used natural branches in all my setups, you don't really need to make your cage furniture sterile and you will never accomplish that anyway. I usually scrub my new branches with a bleach hot water solution (1 part bleach to 10 parts water) just enough to get rid of rotting bark, remove obvious mosses and lichens and surface dirt. Rinse until you can't smell the bleach, let dry. I'd cut off the piece with the fungus, scrub the rest and dry everything. I've never baked anything.