Can a Catholic priest conduct a Catholic baptism upon the request of non-Catholic parents, assuming that they will raise the child Catholic?

The Roman Ritual prescribes

24. An infant of infidel parents may be baptized lawfully even though the parents are opposed, provided that its life is in such danger that one can reasonably foresee it may die before attaining the use of reason. Outside the case of danger of death, it may lawfully be baptized, provided its Catholic rearing is guaranteed, as in the following two cases: (a) if parents or guardians or at least one of them consent; (b) if parents, i.e., father, mother, grandfather, grandmother, or guardians do not exist, or if they have lost their right over the child or are unable to exercise it.

25. Generally, the norms stated in the preceding rubric are to be applied to baptism of infants whose parents belong to a heretical or schismatic sect, or of Catholic parents who have lapsed into apostasy, heresy, or schism.

Note that "Catholic rearing" must be "guaranteed", and it is the priest who must assess that. He may feel that "infidel parents" may not be able to guarantee it sufficiently, even with a fully-qualified sponsor.

The Ritual recognises that the sponsor (or "godparent") will be an important part of the child's religious upbringing:

38. It is the duty of sponsors by reason of their position ever to regard their godchild as a personal charge, and in all that pertains to his Christian upbringing to watch over him faithfully, so that in his whole life he may prove himself true to the promises which they once solemnly spoke for him.

... and it lays down rules which need to be met in order that the sponsor is deemed capable of discharging that responsibility:

33. There should be only one sponsor (who may be of different sex from the one baptized); or at most two may be employed, a man and a woman.

34. To validly act as sponsor it is required:

(a) that the person is baptized, has attained the use of reason, and has the intention of acting in this capacity;

(b) that he does not belong to a heretical or schismatic sect, is not excommunicated whether by condemnatory or declaratory sentence, nor legally infamous, debarred from legal acts, nor a deposed or degraded cleric;

(c) and that the person is not the father, mother, or spouse of the one baptized;

(d) that he is chosen by the one baptized, or by the parents, guardians, or, if these are wanting, by the minister;

(e) that during the act of baptizing the sponsor (or his proxy) physically hold or touch the one baptized, or immediately lift him out of the water, or take him into his arms from the font or from the hands of the minister.

35. To lawfully act as sponsor it is required:

(a) that he has reached the age of fourteen, unless the minister sees fit to admit a younger person for some valid reason;

(b) that he is not excommunicated for a notorious crime, nor excluded from legal acts, nor legally infamous (even though no sentence has been issued to that effect), nor interdicted, nor a public criminal, nor infamous in fact;

(c) that he knows the rudiments of the faith;

(d) that he is neither a novice nor a professed religious, unless necessity urges it and the sponsor has the express permission from at least the local superior;

(e) that he is not in sacred orders, unless he has the express permission of the Ordinary.

36. When in doubt as to whether a person may validly or lawfully be permitted to act as sponsor, the pastor should consult the Ordinary if time allows.

(In every place in the quotes, lawfully is referring to Canon law. If the parents are opposed and a baptism takes place, there may be a civil case to answer.)