240v Split Pase to power 12v Dc fans and lights
The power supply will operate fine from 240VAC. I don't see any reason to involve the neutral. 

You must have a ground for safety, of course. 

Buying a supply makes more sense to me than trying to build a linear regulated supply for several amperes but it's certainly feasible to make a DC supply then regulate it down (with a heatsink or a SMPS module). Mains frequency transformers are large and expensive and heavy for shipping so unless you have an appropriate one in your junk bin it probably doesn't make sense.  

Make sure that metal swarf etc. cannot get into the supply or you'll have a problem and pay attention to the application notes. Also, make sure it's not a counterfeit, for safety reasons. 

Note: If you are asking questions that edge on electrical codes etc. as opposed to circuit design, it's better to indicate what country or countries are involved. From the 120/240 I might assume it's North America.