However that doesn't alter the fact that the nurse could have referred to the patient as first name, last name "and they identify as female"
Why would a doctor need to know this information when having a clinical conversation about catheter care? The nurse was speaking directly to a doctor who simply needed the necessary facts to advise on care.
The patient overheard the conversation and took offence at a description of reality: the reality that the patient is a male who needed catheter care. Most people who hold a belief that contradicts observable science recognise that not everyone holds that belief and that this is OK e.g. Christians who believe in creationism recognise that it's not offensive when scientists talk about the big bang in their earshot.
The way she handled this isn't professional and isn't good care.
I disagree. After the patient overheard her conversation with the doctor, the nurse tried to find a way to adjust how she spoke about the patient in future conversations. From the DM article, this is what she said directly to the patient:
‘I am sorry I cannot refer to you as “her” or “she”, as it’s against my faith and Christian values but I can call you by your name.’
That's a great way of handling it from a dignity perspective. It means the patient gets the "dignity" of not hearing sex-based pronouns again and the nurse gets the "dignity" of not having to compromise her own values. I don't have a Christian faith but I won't compromise my own values either: I won't use someone's preferred pronouns if they differ from their sex because I do not believe that people have a gender identity (nor do I use sex-based pronouns in this scenario, because I'm aware that some people do believe in this and that for some believers it's distressing to hear sex-based pronouns being used). My own approach is very similar to the nurse's. I can well imagine finding myself in a situation where I say something that I subsequently realise may have been perceived as offensive, owing to someone's belief. In that situation, I would do the same as the nurse did and look for a solution that didn't compromise my own values - and in this case, risk compromising patient care (the doctor on the phone needed to know that the patient was male. The patient's belief was irrelevant to that particular conversation).