So they agreed flexible working with you but aren't honouring it?
If you have an agreement to work flexibly, work to it and anytime someone asks you "oh can you just do this tomorrow?" Then it's a firm "sorry, no, I will do it on X day, don't forget I only do 4 days". I always try to focus on providing a solution as I say no. Demonstrates you're not being unreasonable and the business needs will still be met.
I shifted to a similar pattern and you do have to be firm with people. It can also help to manage expectations if a day or two ahead of the day you won't be in you have conversations with people along the lines of "remember I won't be in on X day, so is there anything I'm not aware of you need me to prioritise?"
They agreed flexible working with you so you're not inconveniencing them or failing to meet the business needs. Frankly I grimaced when I saw Boxsets was your first and only reply.
If they want access to you for 5 days could you agree to work from home instead?
Unfortunately I think the only way here is to have calm, firm conversations with people reminding them what was agreed.
Do work understand that HDU is basically intensive care? Sometimes people can be dense.
I'm sorry you're going through this and sorry you're facing losing your dad with all of this added weight in your shoulders.
I don't know if it's any help, but my GP ended up signing me off work when my parent was dying and for a few weeks afterwards. Work just werent getting it (they even asked me if I would be doing overtime in the last weeks of their life!). I appreciate how hard it all is though.