You can use just one surname on your passport even if two appear on the birth cert. The passport form has two sections - name to appear on passport, and name as it appears on the birth cert (if different). This is to allow for exactly the situation you are describing. (Though most frequently it is used by women changing their names on marriage, and by people who wish to have their names appear in Irish on the passport, even though they were registered under the English-version of their surnames at birth).
I imagine that once the child has been registered that way once with the state, then a school should have no problem in accepting that formulation.