Need a thesis extension due to mental illness, but supervisor seems unwilling. How to proceed?
As you say, it sounds like your supervisor does not understand depression at all. Or at the very least, he or she doesn't realise that what you're experiencing is depression and not just typical thesis stress. That's very unfortunate, but all too common (in the general population, not just supervisors). You have every right to an extension.

However, given that you are depressed, you may not be the best judge of your own progress right now. Even students who aren't depressed have difficulty accurately assessing the progress of their research and thesis. Your supervisor has a lot of confidence in you, and seems to think you're "almost done". I'm reading between the lines a bit, but it sounds like your supervisor is a good supervisor apart from being clueless about mental health.

I think it might be worthwhile to have a meeting with your supervisor, and with someone who will be in your corner on the mental health issue (the Honours convenor, perhaps?). The three of you can then put together a plan to finish your thesis. Try to keep an open mind. I don't mean that you should let yourself be pressured into taking on more than you can handle right now. I just mean that there may not be as much work remaining as you thought.