All we can say is that there is no problem with household baptism if the entire household has just been saved. In the New Testament, people were baptized as soon as they were saved. Saul was baptized. The clear, consistent, repetitive teaching of Scripture is that works contribute nothing to saving anyone. Neither baptism nor any other work that we can do has any hope of saving or helping to save us.
Salvation comes through faith alone.
The New Testament knows nothing about an unbaptized believer. Baptism is simply a symbol of what happened to you when you were saved, just as a wedding ring is a symbol that a marriage has taken place. Refusing to be baptized doesn t make one a non- Christian any more than removing a wedding ring ends a marriage. But of course, a faithful spouse is happy, even proud, to wear their wedding ring, just as a faithful Christian will be more than happy to be baptized. Given that new believers were baptized right away, frequently the person who performed the baptism was the person who had just led the new believer to the Lord.