I love the question in the original post. It asks about belief, doubt and the assent of faith.
Purgatory is dogma, and as such it requires the assent of Faith. Most people confuse belief and faith. Faith is one of the Theological Virtues. Faith is not belief. The Catechism defines faith.
1814 Faith is the theological virtue by which we believe in God and believe all that he has said and revealed to us, and that Holy Church proposes for our belief, because he is truth itself.
We make a concious decision to believe, and we do so not because we understand it and not because it makes sense, but because it is revealed truth - revealed to us by Christ through his Holy Church. We gain faith, the habitual virtue of belief without understanding by giving the “Yes of our heart”, by making a concious decision to believe what the Church tells us.
Some complain that we cannot make a concious decision to believe something, as if belief is something which happens to us, but a wonderful example of someone who makes a concious decision to believe something in spite of his doubt is given to us in Scripture.
Then he questioned his father, “How long has this been happening to him?” He replied, “Since childhood. It has often thrown him into fire and into water to kill him. But if you can do anything, have compassion on us and help us.” Jesus said to him, “‘If you can!’ Everything is possible to one who has faith.” Then the boy’s father cried out, “I do believe, help my unbelief!” (Mark 9:21-24)
The boy’s father makes a concious decision to believe. What is amazing however, is the man’s honesty in crying out “Help my unbelief!” He makes a concious decision to believe that which is very difficult to believe - he gives the “Yes of his heart”, but couples it with an honest statement about how difficult it is and a cry to God for help in overcoming his doubt. He doubts because he is human, imperfect, and frail, but he makes a concious decision to believe, and then Jesus cures the boy.
The assent of faith comes first, then light comes and we understand. But some initial doubt is natural. What we do about that doubt is very important. Faith as a virtue is when we habitually believe that which we find difficult to understand, because we trust God’s Holy Church.
The eleven disciples went to Galilee, to the mountain to which Jesus had ordered them. When they saw him, they worshiped, but they doubted. (Matthew 28:16-17)
The disciples worshipped because they believed, but some of them lacked faith so they doubted. Doubt is normal and Chrust understands that, but we have to give the assent of faith, the yes of our hearts, a concious decision to believe first, and cry out to God for help with the rest. Thomas didn’t do that.
So the other disciples said to him, “We have seen the Lord.” But he said to them, “Unless I see the mark of the nails in his hands and put my finger into the nailmarks and put my hand into his side, I will not believe. (John 20:25)
Thomas refused to beleve. Jesus showed him his hands and his feet. Sometimes Jesus does radical things to get our attention. But it he tells us that this is not the way he normally operates, saying,
Have you come to believe because you have seen me?r Blessed are those who have not seen and have believed. (John 20:29)
-Tim-