Any way to mount a ceiling fan using a rectangular "wall" box?
The electrical box should be rated for a ceiling fan and that is not. I would personally add one of these to The surface of the false beam but drill through and screw it into the joist to make sure it will support the weight of the fan. It’s hard to tell if that’s possible from the photos but I think it should be.
1/2 in. Deep 5.9 cu. in. Ceiling Fan Box with Metal Cover
https://www.homedepot.com/p/205383205