We understand that you're going through a period of change and potential stress, Sparklee, as you take in your daughter's older Chihuahuas and try to find a home for the cat. We applause you for trying to find a good home for the cat as in animal shelters the reality of the possibility of death exists. Remember that the Lord cares for all His creatures, including these animals.
God has given us dominion over His creation, which includes caring for its wellbeing. As it is written in Proverbs 12:10 WEB: "A righteous man respects the life of his animal, but the tender mercies of the wicked are cruel."
Firstly, let's address the rehoming of the cat. We pray that the Lord will guide you to the right person or family who will open their hearts and home to this cat, loving it and caring for it as their own. Remember that God knows our needs before we ask (Matthew 6:8 WEB), and He knows the cat's needs as well. Trust in His provision and timing. Pray that someone steps forward that has a Christian believing heart. Remember it is important not to cohabitate with unbelievers or participate in unequal yoking or simply have dealings with someone claiming to be Christian but not holding to Christian beliefs and practices. II Corinthains 6:14 -15.
Regarding the Chihuahuas, first give thanks that God wants us to give thanks in every situation. I Thessalonians 5:18
We pray for your peace of mind and for God's wisdom as you integrate these smaller dogs into your home with your bigger dogs. Pray that your daughters heart be comforted by the fact that you are caring for her pets and that you do care for her situation as she is forced to re-home her pets. Remember Joseph, even when it appeared his brother were cruel and meant harm he cared for them and provided for their needs due to his love of God, His statutes and his fear of God. God rewarded him for that love and obedience. We can pray for all the dogs to be calm, peaceful, and adaptable during this transition. We pray that they will blend well and establish a new pack order that is harmonious. We also pray for your daughter's heart through this time of uncertainty. Her caring heart is a testimony to the heart of God and we pray to remove any fear, sorrow, or anxiety from her about these changing situations. We also pray that she grows in her relationship with Jesus through these times of trial, trouble, testing, tribulations and triumphs as we all must endure. Remember her being cruelly separated as Joseph was is causing you both trouble but God can turn this into great triumph.
The fact you requested to do this in Jesus' name says much about your heart. Remember that by His name and the blood of Jesus we have freedom and access to God.
Romans 5:1-2 (WEB)
5 Therefore, being justified by faith, we have peace with God through our Lord Jesus Christ; 2 through whom we also have our access by faith into this grace in which we stand. We rejoice in hope of the glory of God
You are on that narrow road and you do not neglect to pray in His powerful name.
(Acts 4:12 WEB) "There is salvation in none other, for neither is there any other name under heaven, that is given among men, by which we must be saved!"