I'd just ask "why not" perhaps, but diplomatically, of course...
and if she isn't comfortable conducting an in-office food challenge, would she MIND if you tried it with someone who IS? It sounds as though you are well-situated to find another allergist who would be happy to do an in-office challenge for baked egg.
We value our relationship with our allergist tremendously, and while I might seek a second opinion, I'd never go behind his back to do it for fear of damaging that relationship. KWIM?