Plugging in a two wire corded appliance will not trip a GFCI receptacle with just it's load, it will still need a current path to something grounded to trip, All most all bathroom type appliances come with the corded GFCI and I have never seen a problem. With the way a GFCI works, it cant unless it has a current leakage to ground and that cant happen if the appliance cord don't have one. Both GFCI's will still work if there was a hot leakage that you were to touch while touching something grounded.
Here is some info that you can read to understand how GFCI's funtion: All about GFCI's
