Here we have a very clear example of how uncommon common sense has become in our culture. Our culture’s emphasis is clearly on “Me”—to such an extent that a woman can easily become so blinded by concern for her “fulfillment” that the needs of her children are reduced to an afterthought, if that. But every child that is conceived by the uniting of egg and sperm has a right to a mother and a father—not a mother and a sperm donor.
To pull the conception of a child out of the context of the parents’ mutual love for one another is to sin against God’s design for the human race. To deny context is always a lie. A sperm bank is no match for marital love before and after conception.
God created parenthood principally for the benefit of the child. But our culture puts the cart before the horse so that parenthood becomes a matter of ego reinforcement. This inevitably occurs when we begin to put concern for what we want before concern for what God wants.
The question is not whether the Church would accept such a child into the faith. The question is whether a woman would be so selfish as to deny her child the benefit of a father. Our culture is already plagued by the absence of fathers even when they are physically around, let alone when they are not even known. Since God came first, he comes first. Rather than begin with ourselves, we must always begin with him and make our choices accordingly. This is the only way that ultimately works because this is reality.

While the Church’s judgment concerning in-vitro fertilization treatments may appear cruel and unfair, it is not. Children are a gift, not an entitlement. The Church teaches that

[M]arriage does not confer upon the spouses the right to have a child, but only the right to perform those natural acts which are per se ordered to procreation. A true and proper right to a child would be contrary to the child’s dignity and nature. The child is not an object to which one has a right, nor can he be considered as an object of ownership: rather, a child is a gift, "the supreme gift" (58) and the most gratuitous gift of marriage, and is a living testimony of the mutual giving of his parents. For this reason, the child has the right, as already mentioned, to be the fruit of the specific act of the conjugal love of his parents; and he also has the right to be respected as a person from the moment of his conception. (Instruction on Respect for Human Life 8)

The Catechism of the Catholic Church also reminds us that

Techniques involving only the married couple (homologous artificial insemination and fertilization). . . dissociate the sexual act from the procreative act. The act which brings the child into existence is no longer an act by which two persons give themselves to one another, but one that "entrusts the life and identity of the embryo into the power of doctors and biologists and establishes the domination of technology over the origin and destiny of the human person. Such a relationship of domination is in itself contrary to the dignity and equality that must be common to parents and children." (CCC 2377)

Further reading: Donum Vitae

Homologous artificial insemination and fertilization are generally immoral. The Catechism of the Catholic Church specifically defines and addresses these techniques:

Techniques involving only the married couple (homologous artificial insemination and fertilization) are perhaps less reprehensible [than techniques that require the intrusion of a donor or surrogate] yet remain morally unacceptable. They dissociate the sexual act from the procreative act. The act that brings the child into existence is no longer an act by which two persons give themselves to one another but one that "entrusts the life and identity of the embryo into the power of doctors and biologists and establishes the domination of technology over the origin and destiny of the human person. Such a relationship of domination is in itself contrary to the dignity and equality that must be common to parents and children." (CCC 2377; cf. Donum Vitae 5)

That said, the Church does recognize an exception "for those cases in which the technical means is not a substitute for the conjugal act but serves to facilitate and to help so that the act attains its natural purpose" (DV 6).

So often these days we deal with the argument that a woman has a right to choose to not have a baby—regardless of the fact that it is already in her womb. Now you are suggesting that a woman has a right to have a baby regardless of how the baby is conceived.

The Catholic Church is not attempting to define anyone’s feelings. It is concerned about what God allows and what he doesn’t. Our rights do not originate in how we feel. They come to us from God.

IVF carries with it a host of moral violations. Msgr. Daniel F. Hoye, the general secretary of the United States Catholic Conference and National Conference of Catholic Bishops in 1988, expressed grave concern over the decision by the secretary of health and human services to appoint an Ethics Advisory Board which would approve in vitro fertilization experiments in humans. In a letter he said, in part, "Recent studies suggest that over 95 percent of the embryos outside the mother’s body has made possible the deliberate discarding, freezing and experimental manipulation of human beings at their earliest state of development."

Beyond the horrendous acts that accompany the IVF process, one should recognize that the human person, from the moment of conception, has a dignity that is to be respected. It is not to be used, manipulated, and destroyed so that a woman can have her "right to have a baby." IVF is also destructive to marriage in ways less obvious but just as real (see Instruction on Respect for Human Life in its Origin and on the Dignity of Procreation).

Simply because we want something does not mean that we have a right to it. If we simply acted on our feelings and pursued whatever we felt like having, there would be utter chaos everywhere. We are not the center of creation; God is. It is God who decides what our rights are. God himself designed our sexuality. It is he who has the right to determine how we are to use it, and he has. He designed the context in which children are to be born, nurtured, and challenged on their way to adulthood. Therefore, each child has a right to be the result of God’s design: the product of the love of father and mother as expressed in the mutual self-gift of sexual union, which fulfills the "I do" of their wedding vows. The design is God’s. No one has the right to counter his design by pulling conception out of its context and de-personalizing it.

For those having difficulty conceiving, it can be a tremendous hardship and sorrow. But skirting the moral law is not the answer. There are licit means to enhance fertility or overcome obstacles that inhibit conception. And as you mentioned, there are many children in the world looking for parents to love them through adoption. Trusting God and working within his laws is the way to approach any challenge we are presented with in life.

There is a whole mess of problems with IVF, and some techniques are worse than others. Some, for example, collect the germ cells from the wrong people (i.e., who are not married to each other) or collect them in a morally illicit manner. Some also produce large numbers of children who are either allowed to die or who are frozen indefinitely.

The least objectionable version would be homologous (married-couple) IVF where the germ cells are collected from married parents in a morally licit manner and everything is done to protect the life of the child or children thus conceived. However, even this form of IVF is immoral.

In its instruction Donum vitae, the Congregation for the Doctrine of the Faith (CDF) explains that "the Church remains opposed from the moral point of view to homologous in vitro fertilization. Such fertilization is in itself illicit and in opposition to the dignity of procreation and of the conjugal union, even when everything is done to avoid the death of the human embryo."

The CDF also notes, "Although the manner in which human conception is achieved with IVF and ET [embryo transplant] cannot be approved, every child who comes into the world must in any case be accepted as a living gift of the divine Goodness and must be brought up with love."