I hesitate to point this out to you, but the processional of a Catholic nuptial liturgy is not the Bride’s Grand Entrance but a liturgical processional in which the focus should be on the liturgy and not on the Bride. If that is the case, then the instruments and the music should be appropriate to a Catholic liturgy. Bagpipes are not ordinarily used in Catholic liturgies.
As for the music to be played during the processional, talk with your wedding coordinator or liturgical presider. Some parishes are more lenient than others and permit classical music. Others prefer Catholic processional hymns. Some classical favorites are Wagner’s “Wedding March” and Clarke’s “Trumpet Voluntary.” Possibilities for Catholic processional hymns include “Ave Maria” and “Jesu, Joy of Man’s Desiring.”
If you and your bride want to have a bagpiper at your wedding, have him play before the ceremony begins and/or during the reception.