Our daughter's foreign passport is stuck on a Home Office application. What are the options if we need to travel soon?

We are about to travel from the UK to Greece on holidays in the next 10 days. 5 ½ months ago we submitted our documentation to the Home Office to register our 2 year old as British citizen which included her Greek passport. We‘ve already submitted 2 official requests for her passport to be returned and also send 2 letter with the request reference numbers to the department dealing with our daughters application. We haven’t heard anything back from them as of today.

Our options are limited and none is really ideal:

1. Withdrawing her application would probably trigger a faster return, however this means a cost of approx. £800.
2. Getting emergency documents from the Greek embassy requires us to declare the passport lost which could potentially also harm the application as it will be triggered as cancelled if they check following cancellation.
3. Getting in direct contact with the Home Office is also not possible until 6 full months have passed.
4. Getting emergency documents from the UK could in theory be possible if the Greek embassy denies to issue a passport, but somehow “We will not issue a passport because you already have one” doesn’t strike me as a valid reason.

Do we have any other options? Till now, the best option I can see is to declare her passport lost and travel on emergency travel documents.