The basic federal law about religious hiring is that it is legal for religious organizations (the Title VII religious exemption and the ministerial exception). Does this freedom disappear once a faith-based organization accepts government money? As a general rule, no; faith-based organizations that partner with the government, accepting government funds to provide social services, as a general rule may continue to hire based on religion.
The federal civil rights law that governs federal financial assistance awarded to private organizations—Title VI of the 1964 Civil Rights Act—does not even apply to employment (except if the federal program involved is an employment program, such as an apprenticeship program). In any case, it only bans discrimination in activities and programs on the bases of race, color, or national origin.
However, some specific federal funding programs do ban employment discrimination on various grounds, including religion. But that is not the end of the story. In 1993, Congress adopted the Religious Freedom Restoration Act (RFRA) as a “super” law that applies to all of its other laws (unless the other law specifically denies that RFRA applies—so far, no such “carve out” from RFRA’s coverage has been enacted). RFRA requires that, if some federal law or action is challenged as placing a “substantial burden” on some instance of the exercise of religion (such as the religious hiring freedom), the federal law or action can stand only if the government can prove that the law or action is necessitated by a compelling interest and the government has no way to achieve that vital goal in a way less burdensome on the religious exercise.
Given that RFRA requirement, the U.S. Department of Justice has determined that a faith-based organization that practices religious hiring may nevertheless have the right to participate in a federal funding program that bars employment discrimination. And after Congress in 2013 added to the Violence Against Women Act a ban on religious and other discrimination by grantees, the Obama administration published a FAQ that informs faith-based organizations that, because of RFRA, they may be able to accept VAWA funds even if they hire based on religion.
Sometimes state or local government programs or procurement laws (the laws that govern their grants and contracts) include a ban on employment discrimination without an exemption for religious organizations. Ask officials, ask a knowledgeable lawyer, whether the state has its own Religious Freedom Restoration Act or similar state constitutional principles that will protect religious hiring anyway.