"Testing all things" is the very foundation of scientific inquiry. Why should any aspect of the Bible itself be exempt?
It shouldn't be exempt. I think that one good reason is that neither Jesus nor Paul understood the then Hebrew Scriptures to be exempt.
For example Jesus said this when asked about divorce in Mark 4.
4 They said, "Moses permitted a man to write a certificate of divorce and send her away." 5 "It was because your hearts were hard that Moses wrote you this law," Jesus replied. 6 "But at the beginning of creation God 'made them male and female.' 7 'For this reason a man will leave his father and mother and be united to his wife, 8 and the two will become one flesh.' So they are no longer two, but one.
Jesus doesn't say that God said this, but that "Moses said this", and then He goes back before that to Genesis to show what it was that God wanted.
In Matthew 19 Jesus reaffirms what He had said earlier.
7 "Why then," they asked, "did Moses command that a man give his wife a certificate of divorce and send her away?" 8 Jesus replied, "Moses permitted you to divorce your wives because your hearts were hard. But it was not this way from the beginning.
He is saying that Moses got it wrong. It is obvious that Jesus saw these Scriptures as being written by men not God.
in Matthew 5 he puts it this way.
31 "It has been said, 'Anyone who divorces his wife must give her a certificate of divorce.' 32 But I tell you that anyone who divorces his wife, except for marital unfaithfulness, causes her to become an adulteress, and anyone who marries the divorced woman commits adultery.
In this quote in referring to the Scriptures he merely says that "it has been said".
Jesus did not understand the Scriptures to be read as the literal Word of God. He used His reason, His wisdom and the Holy Spirit in the "Testing of All Things".
Paul did the same. In addition to your quote Paul said this in 1 Corinthians 6:
12 "Everything is permissible for me"--but not everything is beneficial. "Everything is permissible for me"--but I will not be mastered by anything.
Paul is essentially repudiating such things as the food laws and saying that we have to determine what is beneficial to us.
Paul writes this in Romans 2:
13 For it is not those who hear the law who are righteous in God's sight, but it is those who obey the law who will be declared righteous. 14 (Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law, 15 since they show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts now accusing, now even defending them.) 16 This will take place on the day when God will judge men's secrets through Jesus Christ, as my gospel declares.
Our righteousness with God is not based on keeping a set of laws, it is based on our hearts. Do we love selfishly or unselfishly? Is our joy found in self gratification or is it found by bringing and finding joy in others. Is it all about me or is it about God's good creation?
The fundamentalists who insist that the Bible is to be understood as coming word for word from God are modern day Pharisees that pervert the message of God.
Yes, I believe that there is judgement but it is based on our where our heart is, not based on a total denial of our gifts of reason and wisdom and accepting a literal version of the Bible giving us a God who tells us to love our enemy but also that it is fine to slaughter his whole community.
He has told you, O man, what is good ; And what does the LORD require of you But to do justice, to love kindness, And to walk humbly with your God.