Update, I was able to install a round electrical box as a replacement without cutting drywall, but not without another person's help. I could not figure out a way to do it by myself, as I always needed three hands.
The main difficulty was that there needs to be downward force holding the plate that sits across the fan bracket where you mount the box to. This downward force has to be maintained while using one hand to align the box and the other hand installs the screw into the box.
The plate that sits on the fan bracket has six holes in a pattern like this
Code:
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| O O O |
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| O O O |
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The round electric box is mounted using screws that go through the corners of the plate.
Code:
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| X O O |
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| O O X |
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X = mounting screw holes
If you buy a new box with a fan bracket, the supplied screws may not go into the holes in the plate very smoothly and require extra pressure to thread them through the holes. To mitigate this issue, I simply threaded the screws through the holes multiple times in and out and eventually the additional force needs to torq the screws through the plate lessened to a point where they threaded easily. I did it about 10 times for each hole.
Next I cut a long piece of twine about 8 feet long and threaded it through the two middle holes.
Code:
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| X T O |
| | | --------------------------------
| | | bracket
| | | --------------------------------
| O T X |
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T = twine
You can pull on both sides of the twine hanging from the ceiling to provide downward force to keep the plate in place on the fan bracket. The twine is thin enough to squeeze between the edge of the electric box and the drywall. After the box is screwed in, the twine can be simply removed by pulling on one side.
The last thing I did was magnetize one of my phillips screwdriver so I could keep the mounting screw balanced on the screwdriver head since I needed to use one hand to constantly maintain alignment of the holes of the electric box against the holes in the plate.
It still took some time to adjust the plate's location on the fan bracket, rotate and align the electric box holes with the screws, and then have my brother-in-law feed me a screwdriver with the screw and while he then pulls down on the twine. It took about five to six attempts to get everything lined up perfectly and for the screws to take. I did end up throwing all my wife's shoes out of the closet when I dropped a screw a couple times and couldn't find it.
Not sure if you'll ever be in that situation, but it worked for me, and the bonus is that I won't have an ugly drywall patch to deal with (my drywall patching skills aren't great to begin with either, as far as making patches unnoticeable).