I’m not entirely sure the reason for not allowing you to do it under permitted development is correct. You are able to go back 6 m from the rear elevation of the original dwellinghouse under a Notification for Prior Approval for a Proposed Larger Home Extension (as long as no neighbours object). Based on your drawings you are only extending to the rear of the existing three meter extension, so are within 6 m limit for single storey rear extensions. The only potential thing that I could see on your drawing is that the line for the single story extension directly to the rear of the original dwellinghouse goes past the side of the original dwellinghouse by a tiny amount (though this might just be the drawing). If it is the case that the proposed extension goes past the side of the original dwellinghouse by even a millimetre then it would be classed as a side extension and therefore the total width of the extension cannot be greater than half the width of the original dwellinghouse, that may be why they are not allowing PD, however it might be that the drawing is a little bit off and that’s not the case at all.
In terms of ‘inappropriate development’, as you know that is a very subjective term and basically is what the planner determining the application and the manager who signs off think (though the planner and their manager often come to different conclusions themselves, but obviously the manager’s opinion appears to be final in most cases).
Obviously what is inappropriate to one person may be appropriate to another. I don’t know what materials your extension is made up of etc. But the inspector that would determine the appeal is just another person with their own opinion, and I have seen countless times where the inspector has come to a different opinion to the planner.
It would be worth seeing the actual report written by the planner and some elevation drawings of your proposal, including proposed materials so that I can give you my opinion, though again it is just another opinion to throw into the mix.
With regard to the new houses, it probably won’t influence the Local Planning Authority, but if you appeal it you don’t need to worry about what the local planning authority think, just the inspector!