Here is an important distinction to keep in mind that will be helpful to you in your dilemma:
Catholics believe that the man and woman give themselves to each other in marriage. The reason why the question "Who gives this woman to this man?" never appears in a Catholic marriage liturgy is because the freedom of both parties to marry each other is so important that any suggestion that there is a lack of freedom by the bride to enter into the marriage -- that she has, instead, been "given" into marriage by her father alone or her parents together -- could call into question the validity of the marriage. I strongly urge you and the bride's mother to stoutly refuse to participate in "giving away the bride" on the basis of religious principle.
Now you may be wondering about the fact that a father does often escort his daughter down the aisle in a Catholic marriage liturgy. The reason this could cause you to wonder is because very often people conflate the act of escorting a woman down the aisle with the act of "giving away the bride." These are, in fact, two separate and distinct rituals. Ordinarily, Catholic parents are free to escort their daughter down the aisle to meet her intended groom without involving themselves in a non-Catholic ritual of "giving away the bride" that seriously conflicts with the Catholic understanding of marriage.
In your case though, you must determine if your daughter's marriage will be presumptively valid. If she is a Catholic bound by Catholic marital law, then she needs to be working with a Catholic priest or deacon to ensure that her marriage in a non-Catholic ritual will conform to Catholic canon law. If she does not, then her marriage will be presumptively invalid. If the marriage is presumptively invalid, then I cannot recommend that you engage in any marriage rituals that will give the appearance that you approve of her choice to flout her Catholic faith.
What are the rules for attending weddings?