When that happens it generally means that an organisation has not responded in time and that is putting pressure on the timescales for publishing an article that they want to do.
You do not respond to the pressure no matter how “embarrassing” they say they are going to make it for the organisation.
Public sector bodies are far from perfect and do not rely on them to help meet your timescales.
That applies to both FOI and data protection.
Also try and ignore any pressure from internal communication teams – if they think they can do any better dealing with FOI requests, transfer responsibility to them.