Does anyone know how to split the 240 V wire components of a dryer from the 120 V?
Oh no! You can't do that hack-a-dack.

That transformer probably has what you need, but you need to draw the 120/240 power in a very particular way.  You cannot just slapdash a 120V plug and call it good.  If you do, there's a 50/50 chance that you'll plug it in and BANG!  

The professional installer who installed that transformer, or an electrician of equal skill, will know the correct way to extract 120/240V from that transformer.  Call him and have him install a NEMA 14-30 receptacle with appropriate circuit protection. 

Hacking the dryer to have totally separate power systems for heater and motor isn't outside the realm, but it's going to require a fair quantity of additional parts, because the parts you need are not in the dryer and won't be available on a 240V island.  Notably, 2-pole relays where the dryer now uses 1-pole.  The cost and time delay to source those parts, iteratively, is probably going to be prohibitive.