Babydoll
They are talking rubbish but this does not help you in the short term.
If your claim date is 4 July and a completed PIP1 is not available at this time then your DLA payments will be suspended and you will be given a new date for a claim to be made by which will give you a further 4 weeks. If the claim is not started by this second date then your DLA claim will be closed, if you do start a PIP claim then any held payments will be released.
Even if the PIP1 arrived tomorrow and this seems very unlikely, then you would struggle to get it returned by the 4th so I would look at your options for making the claim over the phone.
As a separate issue I would make a formal complaint to the DWP referencing the Equality Act and also contact your MP about the situation.
Complaining to the DWP Contacting your MP Gordon