and what scriptural proof do you have for such a claim?
The Gospels.
The Pharisees knew that Jesus existed; they saw him, heard him and saw his miracles. They, mostly, did not believe, said that he did his miracles by the power of the devil and tried to trap, and then kill, him on more than one occasion.
Herod and Pontius Pilate knew that Jesus existed, saw him and talked with him; they didn't believe.
The demons that Jesus drove out of people, not only saw him but were well aware who he was - they didn't believe and become his followers.
The devil himself knew who Jesus was.
"Believe on the Lord Jesus and be saved" is what the Apostles said, NOTHING else. Where in Acts 16 did the Apostles tell the jailer when he asked "What must I do to be saved?" did the Apostles say, "well, first off, you must believe that Jesus is God, who died for you!"?
Have you read the next verse; "then they spoke the word of the Lord to him", Acts of the Apostles 16:32?
It is not impossible that the jailer heard Paul and Silas singing hymns and praying, Acts of the Apostles 16:25; he might even have known the reason that the two were in jail. Then he saw God's power, in the earthquake which set the apostles free.
Paul told him to believe in Jesus, and then explained who Jesus was.
Ever hear of death bed conversion? While I don't, many Christian do.
Of course I've heard of it.
They're lying there dying and they accept Jesus as their Lord and Savior and automatically go to heaven when they take their last breath.
i) If they're Christians, they are already saved and don't need to accept Jesus again.
ii) Such a conversion would not come from nowhere; the person may have been in hospital for a while, been visited by the chaplain, had time to accept that they were dying, talked to other Christians etc. They may have heard the Gospel earlier in their lives.
When was there time to accept Jesus as God? When even the Church Fathers took some 400 years to develop the Trinity doctrine???
It would depend on how the Good News was explained to them.
If someone said, "Our sins separate us from God and without God there is no hope. But God himself came to live on earth as a man and experienced our lives, our temptations, pain, loneliness etc. This man was Jesus who lived a perfect life and then died on a cross to reconcile us to God and grant us forgiveness for all our sins."
A person could believe that, without necessarily completely understanding it. They wouldn't need to know, and understand, the doctrine of the Trinity to be saved.
It doesn't matter how long the church fathers took to develop the doctrine of the Trinity; Scripture says that Jesus was God, was with God before the creation of the world and was, and is, one with God.