After all this trouble – minimum 4 weeks lost – I replaced THTTPClient with Indy TidHTTP. Not too much work and the basics worked within an hour or two.
One new problem introduced by Indy was that TidMultiPartFormData did not work as expected. For a quick solution wrote data in a TStringStream and sent instead.
SideEffect: The app did not work when compiled in 10.1 Berlin. Now it does !