YABU
They clearly don’t agree you can do your job adequately in 2 days and you clearly did not convince them otherwise at your meeting. They are totally justified in refusing your request if they can reasonably conclude that your request would be detrimental to the functioning of the business/ institution/ organisation.
Your employers duty of care extends to considering your application and giving you their reasoning for declining it, if they do so. I appreciate you are stressed and feel you are in a difficult situation- you haven’t said what it is- but they are not necessarily failing in their duty of care because they are not giving you what you feel would work for you. They have considered your application, has a meeting with you and so it’s not clear in what way you feel they have failed- other than the fact they did not agree with your assessment that your job could be done in 2/5ths of the time you currently work.
I’d be very surprised if you are correct you can do your job, in its entirety and to a high standard, in 2 days when you currently work 5 days. I’d be really surprised that a head of department could be so part-time and still fulfill the requirements of their role. Who makes decisions that need a HoD level sign off of you are not there 3/5ths of the working week, for instance? That seems unworkable and leaves your team very unsupported.
The fact that you cannot find a job elsewhere at your level that is only 2 days per week would tend to support your employers stance, in all honesty.
I think you are also putting yourself in an incredibly vulnerable position but telling your employer that your duties and responsibilities can be fully, satisfactorily completed in 2/5 of the time you use currently. That rather calls into question what you are doing now- you are clearly underworked and must spend a lot of time twiddling your thumbs. If true, your employer may decide that your role as it stands is obsolete and the department/the business could be reorganised to abolish your role and reallocate parts elsewhere or achieved in a different way. If you’ve told them you only need 2/5ths of your hours to do manage what you do currently, I wouldn’t be too surprised if they start looking into whether what you have requested is accurate.
You can be excellent at your job, your employer could think a great deal of you and be prepared to make a lot of concessions to keep you- but very, very few people are 100% irreplaceable. Your employer has clearly shown you today that they will not do anything to keep you. There is a limit- this is clearly it.
your best bet, I think , is to speak with your employer about what level of reduced hours they would consider. It could be on a temporary basis, to see whether it works for both parties. But you’d have to be prepared with what you might do if it does not work and they need you back full time. They may not wish to consider temporary changes, in case you then try to argue it set a precedent.