This topic explains that the passport number of the professional referee can be omitted from the application form.
First this caseworking study seems outdated, I can't find it online.No. The caseworking instructions say:
6.3.7.9 Those acting in a professional capacity may not wish to provide their personal details. In such cases, we would be content for the person acting as a referee to put his or her business contact details. In addition, if the person does not wish their passport details to be seen by the person applying for citizenship, they can omit this from the form, entering “details available on request”.
Secondly the current online form forces you to fill a date. Would you just input an obviously wrong birth date (like 01/01/1920) and put a note at the end clarifying your choice?
Or would you go ahead and just overcome the embarrassment with that coworker. I suppose date of birth isn't a piece of info that people should be reluctant to provide especially nowadays in our data and social media era.