Tricky patch in ceiling of closet
I found a hole in the ceiling of our hallway closet in our townhome. (See picture, taken from below). It is 14" x 2", and there appears to be wood (joists?) right up to the edge of the hole on both small ends and one of the long sides (the bottom on the provided image). I assume the exact placement of this hole between these joists has something to do with why this hole is even here to begin with, but I have no idea. The other long side butts right against the cabinet frame, and on the other side of that is our HVAC vent, which you can see at the top of the photo. Most drywall patching videos I've seen suggest to add a backing board, which I can't do here because of all the things around the hole. I considered doing a "California patch", but it looks like this hole might be too big for that, especially since it is in the ceiling. I know there are metal mesh patches as well, but similarly, I'm not sure if that is appropriate for this sized hole.
Bits of stuff have been falling from the hole recently into the things in our closet, so I'd like to patch it up somehow. Wondering what the simplest solution is here. I'd love to not have to make the hole bigger, if possible. It's tricky getting tools in here because the closet has shelves. Thanks for any advice.