First off, the church has never defined anything about the nature of purgatory, so the concept of someone "burning in purgatory" is not de fide dogma. (I'll grant you that a case can be made for it from Catholic theological speculation or popular piety.)
But I think you need to go back to your Bible, and reflect on what Paul meant when he spoke of someone being "saved, but only as through fire," what Hebrews means when it says "Our God is a consuming fire," and what Revelation means when it says "nothing unclean shall enter" heaven.
Sounds like the dross has to be removed in order to enter the presence of an utterly holy God.
It means either God doesn’t know how to forgive, or He only forgives part-way.
Quite to the contrary. It means he forgives completely, so completely he's going to cleanse every bit of the stain of sin from your soul. Not merely the juridical guilt, but even the damage sin did to your soul and the attachment you may have to it.
And that might hurt a little.
You refer to 1 Cor 3...which reads:
” 10 According to the grace of God given to me, like a skilled master builder I laid a foundation, and someone else is building upon it. Let each one take care how he builds upon it. 11For no one can lay a foundation other than that which is laid, which is Jesus Christ. 12Now if anyone builds on the foundation with gold, silver, precious stones, wood, hay, straw— 13 each one’s work will become manifest, for the Day will disclose it, because it will be revealed by fire, and the fire will test what sort of work each one has done. 14If the work that anyone has built on the foundation survives, he will receive a reward. 15If anyone’s work is burned up, he will suffer loss, though he himself will be saved, but only as through fire.
16 Do you not know that you are God’s temple and that God’s Spirit dwells in you? 17If anyone destroys God’s temple, God will destroy him. For God’s temple is holy, and you are that temple.”
Pretty tough to pull Purgatory out of that!