Is there any hope for this power supply to work on 220V rather than 110V?
Simple answer: NO.


  There is also a somewhat large capacitor that says 200V 270uF.


The capacitor is for 200V, which is sufficient for a fully rectified ~120V. 
~230-240 would burn it with its 340DC. ~230-240V tend to use 400V capacitors.

I don't quite understand the triac part and it does requires AC to function. I suppose it's a heating element and in such a case, you cannot use 240V w/o a transformer.
Otherwise, not recommended but if it's a low powered switch power supply, you can cut half of the wave of 230-240V via a single diode and feed DC to the full wave rectifier. Again not recommended and relatively dangerous.