Its been a very long while since ive seen the old Adam West series, so I dont completely remember all these characters. From what i remember on King Tut, he was a relatively good guy who went a little crazy and started believing he was an ancient being of great power who deserved to rule the world. Thats pretty much the same story they did with Maxie Zeus. Why bring in Tut to do the same thing?
As for Egghead, I remember nothing about the guy other then i think he was played by Vincent Price. So lets look at what made Egghead "Egghead". A fascination with eggs, which you said should be eliminated. A weird bald look, which would probably have to be toned down. Played distinctively by Vincent Price, who obviously couldnt reprise the role. After that all we get is a generic mad scientist. Batman:TAS already had several. Why bother attaching the silly Egghead name to them when in the end they'd really have nothing to do with the Egghead character?