The third pin on the US plug is the ground. If your converter doesn't provide a grounded plugin you could use something like
this to convert the grounded plug to un-grounded, then into your converter.
This removes the ground from the wiring completely and is not recommended for safety reasons. If your unit were to short circuit there will be nowhere for the electricity to ground to....other than possibly through you.
I need to be clear that I am not an electrician and would probably get a converter that accommodates grounded US plugs. With that said, using an adapter to remove the ground from the US plug will work.
To your point, if your converter accepts 2 prong Euro plugs at 110v output...your solution should work...albeit, you're removing the ground either way, which does cause a safety concern to me.