Speaking from (too much!) experience, just say whatever will keep him happy - it won’t be any good using reason or logic.
Don’t worry if it’s blatantly not true! In dementia carers’ circles they’re called ‘love lies’. So as regards the death certificate., I’d just say e.g., Yes, I sent off for that yesterday - or whatever is appropriate.
If his short term memory is very poor, he’ll probably forget whatever you say very quickly, so you can rinse and repeat.
Among other things, my mother went through a long phase of being convinced that her sister had ‘stolen’ their mother’s house (!) and would get really worked up about it.
Once I’d finally realised that even a signed and sealed guarantee from the Lord Chancellor wouldn’t have convinced her, I just started saying e.g., ‘Dear me, that’s terrible - I had no idea! I’ll get on to the police/a solicitor first thing tomorrow.’
It always kept her happy, or at least not fretting about it.
I second a pp’s advice about the Alzheimer’s Society Talking Point forum. It was a lifeline for me when things were very difficult. There aren’t often any easy answers, but whatever you’re going through, someone will have been there - and at least you can let off steam to others who know exactly what it’s like.
People who haven’t lived with dementia so rarely do. (But all too often they think they do!) ?