Update :-
Thanks for all the feedback so far … I've been playing some more and there might be more to it than simply a battery/adaptor problem.
Seems like there might be a heat/range/interference issue in the mix as well.
Indoors, over a 5-10ft range, all is good irrespective of configuration.
Outdoors, in the garden in bright hot sunshine, the battery only configuration is still fairly reliable, but as soon as I connect the mains adaptor (with a mains extension lead), with or without a battery in the camera, the connection drops very regularly, as though either the adaptor is acting as an antenna or causing the wi-fi to be more sensitive and so being affected by external interference … or maybe the camera is simply shutting down in the hot sunshine.
So, as it stands, provided I don't get too far away from the camera (what is the range of a good wi-fi connection?) and don't try to run the camera off the mains, I can get it to work … not quite what I had in mind
One configuration I've not tried, having only just thought of it, is to wedge the "dummy" battery, as used in the K-70, into the KP and run the mains adaptor through that … something for next time