Yes, HT's discretion is deliberately undefined, because there is no way that all possible exceptional circumstances are codified. Individual local consideration is considerably better
If a HT decided not to authorise, it's on them. Regardless of whether they have abandoned their right, set out in law, to be the one who decided (in favour of eg the LA, as described in some posts), or to governors or to anyone else.
The HT does not however recommend a fine, they simply decide that what is or isn't authorised. The LA fines based on their fining policy and what is entered in the register.
Compassionate visits are usually day or two, but it sounds as though you were away for over two weeks.
It is utterly shite when your school does not believe you - ask any parent of a genuinely chronically ill DC who is now in a climate of disbelief because of the number of people who think it's OK to call in sick.
You have done the right thing in being truthful with the school, and as you say you have proof of expected death, it might be worth a brief email to the school attaching an image of that. Because although he has pulled through this time, presumably there's going to be a next time and you will want your DC present then as well. Even if you cannot overturn the HT's decision not to authorise, I'd get as much recorded in writing as possible, in case it's needed in future