Any way to mount a ceiling fan using a rectangular "wall" box?

I was replacing a ceiling fan in a newly purchased home, and I discovered that the previous fan was mounted to a rectangular "wall" box. The box is attached laterally to a ceiling joist in a converted garage. There is some trim wood to give the ceiling joist an exposed beam look, and I do not believe that there is enough space for a standard ceiling box. The joist is not centered with respect to the trim, so mounting directly beneath the joist is not ideal.

I attempted to use the existing box, but the threading stripped as I put in the second screw. Now I’m questioning whether this box was even appropriate in the first place.

Is there any way to mount the fan to a box that will fit in this space? My first thought is to replace the box and add support via a brace to the joist or by securing a piece of wood above both the joist and the box.

If mounting to the box itself won’t work, can I secure the fan to the joist via a bracket or supporting wood, and simply house the wiring in an adjacent rectangular box?

The location is Texas, USA.

View of the electrical box from above the joist.

View of the fan mounting hardware below the cosmetic wood.