What do we when outside government authorities are involved in our child raising?
This becomes a difficult question to answer well because much depends on the reason the government has seen fit to interfere. The best that I can do is answer in general and hope that I have hit upon your particular situation.
The apostle Paul tells us, "Let every soul be subject to the governing authorities. For there is no authority except from God, and the authorities that exist are appointed by God. Therefore whoever resists the authority resists the ordinance of God, and those who resist will bring judgment on themselves. For rulers are not a terror to good works, but to evil. Do you want to be unafraid of the authority? Do what is good, and you will have praise from the same. For he is God's minister to you for good. But if you do evil, be afraid; for he does not bear the sword in vain; for he is God's minister, an avenger to execute wrath on him who practices evil. Therefore you must be subject, not only because of wrath but also for conscience' sake" (Romans 13:1-5). The general rule is that governments have little reason to interfere with the lives of the righteous and so the righteous are instructed to cooperate as best that they can. It is especially telling that Paul wrote this shortly before his own arrest in Jerusalem. The Roman government of that time did not hold a favorable view of Christianity.
Peter also argued the same, "Therefore submit yourselves to every ordinance of man for the Lord's sake, whether to the king as supreme, or to governors, as to those who are sent by him for the punishment of evildoers and for the praise of those who do good. For this is the will of God, that by doing good you may put to silence the ignorance of foolish men -- as free, yet not using liberty as a cloak for vice, but as bondservants of God" (I Peter 2:13-16). Hence, the general rule is that Christians cooperate with the governing authorities as best they can.
Of course, this will eventually lead to a conflict of interest. Corrupt people exist throughout the world, and governments are not immune from their presence. In fact, because governments are centers of power, they tend to accumulate more than their fair share of the corrupt. "Moreover I saw under the sun: In the place of judgment, wickedness was there; and in the place of righteousness, iniquity was there" (Ecclesiastes 3:16). We must keep in mind that governments exist by the ordinance of God -- whether they accept God or not. When there is a conflict between the government of men and the law of God, God's law always prevails as it is the higher authority. Hence, when Peter and the apostles were commanded by the ruling council of their day to stop preaching about Jesus, their response was, "We ought to obey God rather than men" (Acts 5:29).
If there is a specific issue that you would like to see addressed, I will be happy to do in a follow-on response.