How do I dig a fence post hole in a cramped corner?
I don't think there is any clever way to accomplish this that is going to make it any less unpleasant. It's a tight space and it doesn't sound like there's any hope of getting an auger in there. I'd get a nice sharp (yes, sharp) post hole digger and a spade, plus a wheelbarrow to haul the dirt out, and have at it. Really, digging a 3-4' hole there isn't going to be that awful.

Only thing I can think of that might make it easier is to try and convince your neighbors to temporarily take down part of their fences to clear the area and give you more area to work in. You might be able to grease the skids a bit if you pitch the new post as something that all of the fences could then attach to for improved support. Normal people aren't going to turn down your offer to improve their property for free, and the less work they have to do, the more true that becomes.

I'd actually like to see a photo of the current situation. 4 fences meeting like that with no solid support sounds really janky. I can't imagine anyone is particularly happy with it as-is.