The eligible student may be eligible for Federal student aid in accordance with subparagraph (B) for the remaining cost of attendance amount for which the student is responsible. (ii) Private institution (I) In general An eligible institution that is a private, nonprofit institution of higher education shall— (aa) with respect to an eligible student described in subclause (II), cover excess costs toward meeting an eligible student’s cost of attendance in an amount equal to or greater than the amount of the grant award to the eligible student described in paragraph (3)