<landlord, welcome to my life
As someone who puts time as priority number one, and has learned all the little time saving tricks over the years, there is no filler you can put in there without a mesh or a backing. If I was making this thread, I wouldn't believe me, and I would try it on my own as it seems so simple. It will likely work but it will look like ****, you will notice it constantly and you will always wish you took another route.
The correct way, and believe it or not when all said and done might be the easiest way. You won't believe me until you try it, I know it lol. Is to cut out a section of the sheetrock as wide as the joist and just put a new piece in.
The way that might be less time consuming, is to take a piece of aluminum mesh for hole patching, cut out a circular hole the size of your can, carefully making sure not to bend the piece at all, and then apply it to the hole, then compound over the mesh.
http://www.amazon.com/gp/product/B000BQSEQA/?tag=atomicindus08-20
Or - an idea I just came up with because of your particular situation. Take the cover off, put some compound in there, put something flat like a shim against it so the compound doesn't droop, put the cover back on so that springs of the cover hold the shim on. That's assuming the cover attaches with springs. Cover the shim in wax paper or something so compound doesn't stick to it. This should give you a flat enough surface to apply a skim coat on afterwards. Just be careful sanding as it won't be very strong.
Also if you use compound just for the purpose of repair, I prefer the self mix quick dry stuff. Not because it's quick dry though. It's a different compound from the pre-mixed kind and is considerably stronger and more resilient. It also does well with water prone areas because it doesn't not absorb water like regular compound. Regular compound waits for the water evaporate to become hard, quick dry compound goes through a chemical change with the water, similar to a cement. After the chemical change has taken place, it's not going to react to water again. I mean it's easy to figure this out just when you go to clean your tools after the stuff has dried. Fully dried regular compound will just turn soft and comes off easily with water, the quick dry compound stays pretty hard.