Apart from believing that's part of SOX's "charm", it would also better imitate the flame of a gaslamp. Gaslight via a mantle is quite different from SOX or any other electric lightsource and is surprisingly tricky to imitate well. As it is a point source of light, a large amount of light would not be expected except by providing more mantles.The colour encountered is a pale straw / lime green which is at variance even with candle light, although both forms are organic in nature. Tilley lamps although running on paraffin produce the same colour light as they also use mantles. Colour rendering is still distorted to a degree, with most colours dulled. SOX by comparison has a very marked distortion as has already been discussed.
That said, I think SOX is a worthwhile conversion and the gear should be easy to obtain. The secret is to either keep it discreet via "cut off" distribution (ie the lamp is shielded from view unless you are pretty much under it) or you go the opposite direction - everything out in the open and stuff the purists! That said, if the lamp is to be hung vertically, some form of ring refraction may be useful.
The use of aluminium as a reflector has already been suggested to reduce skyglow, however I suspect you needn't worry too much about that here for an 18W lamp, however there is nothing wrong with a little "discipline" to keep the light where you want it most. Bear in mind that many gas lanterns were traditionally fitted with opal glass on the upper panels, often to hide the mantle support etc.
Whilst aluminium has the benefit of being light and easy to work, the lack of upward light will take away the lantern's heritage or worse still make it look like just another reproduction. Simple opal glazing would act as a reflector yet still allow some upward transmission. By designing the panels to work with the existing structure, they could be slotted or glued in place, whilst retaining the exterior glazing as fitted.
One of the opal panels would be drilled to accept a suitable bracket for the BC lampholder, whilst its neighbour opposite would have a lamp support attached. The support could alternatively be fitted on the horizontal panel in the top as per the diagrams. As has been already suggested by others, small gauge wiring would be ideal to hide within the lantern structure.