Make sure you have the diode completely pressed into the module. It should be recessed into the module a little bit when it is pressed in all the way. Bend the pins a little bit, only wide enough to let the driver bard sit flush against the module. bend the pins back down to the board and in the direction of the pads that they need to be soldered too. Solder and short the appropriate pins making sure the driver is parallel to the module so it won't hit the sides. If you get it on perfectly straight you probably don't need any tape to protect from shorts but if you think you'll get some metal on metal contact anywhere wrap it in heat shrink or electrical tape.
As long as you have the driver board pressed all the way up against the module it should fit.
Also, that barrel screw on part is not necessary, if you can't fit it over the driver just leave it off.