-
First thoughts are it's a bad idea to cut through the stile and panel. I normally never suggest changing a door but in this case if you kept the original one and got one purpose made to contain a catflap you could at least change it back easily if you needed to.
A picture would help to assess the type of stile/rail/panel construction then it might be possible to give better advise. I'm guessing if you really need to go ahead with it sticking a thin panel to the butchered door with epoxy resin would allow you to even out the difference in thickness between rail and panel and fill the gap left by the moulding.
Installing a cat flap in a front door does sound like it would be a security nightmare.
Can’t you just give the cat a set of keys?