You will need to use transform warp on the bump map to bring it into shape
, also change the UV properties to V/Y Tile 0.5 - so you don't get two pull rings
If you need it to cross the centre point - then the better way is to create an additional model with the circle as a front surface then put the bump map on that (use combine 3D layers to bring it together with the other model). You will not need to distort it that way.
Thank you Dave!
I solved the problem using the second technique you explained, since I needed to cross the center point. I think it would be better if Adobe make it work in the bend x 360. There should be a way in which Photoshop "un-distort" the map for you to place your bump image and then "re-distort" it for it to accommodate in the extrusion surface.
This is the final result using the frontal extrusion anyway.