Sorry, all we had to go on is a somewhat vague description of what you are trying to do. A picture paints 1000 words and all that.Through what wall, if the doorbell is in the frame there is no option to go through the wall.
If I mounted the doorbell on the wall you wouldn't see my door and the bell would be awkward to reach.
I'm not sure what would look naff, you wouldn't see any cable at all.
I get you might not see You might not see the cable but you'd see the additional trim/trunking that otherwise wouldn't be there.
How are you dealing with it on the outside, e.g. how are you getting the able into the camera from between the frame and the rear entry of the camera which is in the middle of it?
Technically yes, realistically no. Do you honestly think someone is going to come and cut the cable, put an RJ45 on the end of it and tap into your local network?Is running a network cable outside your house in a fairly accessible location possibly a security risk? I mean, they would have to be determined and clued up, but I’m sort of unclear why a wired Ethernet doorbell was thought of as a great idea from a company who does security solutions. Cameras are similar but I guess usually out of reach.
They also wouldn't expect you to run the cable for the doorbell externally, its rear entry only so you need to make a hole in whatever you attaching it to.