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You are totally right. Allowing him to stay in the house with you and your children sets a bad example. It shows approval of his adultery, not only for your children but for other relatives and neighbors. Remember you, as a Christian, have the Holy Spirit living inside you. Does He want to be put in this situation? No!
2 Corinthians 6:15 What harmony is there between Christ and Belial? Or what does a believer have in common with an unbeliever?
Your desire to make a stand for righteousness in your life is commendable and God will reward you for it. Adultery is the ultimate abuse to a spouse. They have replaced love and dignity (the state or quality of being worthy of honor or respect) with lust, shame, disgrace, humiliation and loss of self-respect. I say this because of this scripture.
1 Corinthians 6:18 Shun immorality and all sexual looseness [flee from impurity in thought, word, or deed]. Any other sin which a man commits is one outside the body, but he who commits sexual immorality sins against his own body. (Amplified Bible)
Satan knows that even though adultery degrades a person on the outside, as other sins do, it's the only sin that degrades a person inside. It gives him power over a person's soul. The soul of a person is their heart (the area of desire and choice), mind (thoughts, mental evaluations) and emotions (feelings and expressions). Satan separates the spirit of that person from God's spirit (which connected at the time of salvation) and re-connects it with Satan's spirit, which is the sinful nature that we are all born with (due to the fall of Adam).
When sin of that magnitude moves into the soul of a person and is not removed immediately, through repentance and a changed heart, it increases Satan's ability to control that person in a greater way than with all other types of demonic influences. Whatever strength that a person may have in their heart or mind to resist sin and temptations deteriorates quickly; rendering the Holy Spirit helpless to intervene for them when they are in trouble.
He has treated you in a violent manner by giving away what was cherished by you.
1 Corinthians 7:4 The wife does not have authority over her own body but yields it to her husband. In the same way, the husband does not have authority over his own body but yields it to his wife.
He has mistreated and disrespected you and now wants to continue by sleeping in your home. His conscience has been seared by his sin.
How Should A Wife Or Husband Respond To An Adulterous Spouse?
Any person who desires to re-enter into a marriage contract (which him sleeping in your home would appear to others to be) where the adulterous spouse has decided to continue in their adultery, is going against God's word.
1 Thessalonians 5:22 Abstain from all appearance of evil.
2 Corinthians 6:14-18 Do not be yoked together with unbelievers. For what do righteousness and wickedness have in common? Or what fellowship can light have with darkness? What harmony is there between Christ and Belial? What does a believer have in common with an unbeliever? What agreement is there between the temple of God and idols? For we are the temple of the living God. As God has said: "I will live with them and walk among them, and I will be their God, and they will be my people. "Therefore come out from them and be separate, says the Lord. Touch no unclean thing, and I will receive you. I will be a Father to you, and you will be my sons and daughters, says the Lord Almighty."
God is extremely strong about how we conduct our lives since we are His temple. No Christian is ever to subject themselves to the abuse of an adulterer!
1 Corinthians 6:19-20 Do you not know that your body is a temple of the Holy Spirit, who is in you, whom you have received from God? You are not your own; you were bought at a price. Therefore honor God with your body.
Your body is how you think, speak and act. Everything we do should be in accordance with God's Holy Word. It's good for us to have his Bible as a measuring stick for the situations of life.
1 Corinthians 3:16-17 Don't you know that you yourselves are God's temple and that God's Spirit lives in you? If anyone destroys God's temple, God will destroy him; for God's temple is sacred, and you are that temple.
Scripturally from studying the marriage and divorce laws in the Bible, adultery is the only act that totally sets a spouse free. There is no longer a contract of marriage when the vows are broken. The injured spouse is fee, even to remarry if they choose.
Matthew 5:32 But I tell you that anyone who divorces his wife, except for sexual immorality, makes her the victim of adultery, and anyone who marries a divorced woman commits adultery.
To give him a choice to leave or come back is fine. However, if he comes back you must insist that he base in on God's Word not just feelings for you and the children. Unless he repents and his nature is changed to a godly nature the lustful spirits that took him over will only hide for a little while. I am praying for you and believe God will bless you. Thank you for writing.