Replacing GFCI Plug on Hairdryer
The main difference between your old GFCI plug and the new one you propose is that 2 wire only GFCI plugs lack what is called a grounded neutral detection circuit -- this circuit helps improve the GFCI's sensitivity to neutral-to-ground faults.  This grounded neutral detection is unnecessary but harmless when the load is ungrounded -- your situation is no different electrically than a 2 prong double-insulated appliance plugged into a conventional GFCI receptacle.