... I didn't check your link first... the adjustable one you linked isn't rated for rotation. That is what differs between boxes... and to clean this long message up because I'm tired and rambling. I can't tell if the box you have in the wall has an adjustable bar. I have to assume its screwed into a board that runs joist to joist. And that that board has at least 2 screws on each end
First option. Cut drywall in a square, keeping drywall intact. Replace/relocate existing items. Remount drywall.
Make those screws come off and remount a shallow box, assuming your fan is light enough.
Use something like a dremel to shave down the lip of the box, not the screw holes do to structural support. (The screws need a certain depth to anchor well)
Directions for DIY 2x4 fan brace:
-2x4 to span the distance between the joists
-six 3 inch nails or screws for "toe nailing"
-the new work cieling fan fixture box
-roughly six 1 inch nails or screws to fit the holes of the new work box (or however many it takes)
-possibly a razor knife to cut insulation
-drill, bit for screws if using screws, and drill bit for pilot holes
1. Cut insulation and separate it above the hole.
2. Pre-fit 2x4 to make sure it fits snugly, not not to tight.
3. Drill 2 pilot holes at both ends of 2x4 at 45⁰ angle and a 3rd hole on the opposite side between those so you have 3 holes at either end, 6 total.
4. Seat 2x4 flush on top of drywall beside existing hole.
5. Screw in 3 inch screws.
6. Seat new work box against the 2x4 and in the ceiling hole. So its flush against the 2x4.
7. Put in 1 inch screws.
8. Put wires back through box.
9. Mount fan