Scripture is clear as to what every single Christian will experience at the Bema Seat of Christ...
1 Corinthians 3:9-15 New International Version (NIV)9 For we are co-workers in God’s service; you are God’s field, God’s building.
10 By the grace God has given me, I laid a foundation as a wise builder, and someone else is building on it. But each one should build with care. 11 For no one can lay any foundation other than the one already laid, which is Jesus Christ. 12 If anyone builds on this foundation using gold, silver, costly stones, wood, hay or straw, 13 their work will be shown for what it is, because the Day will bring it to light. It will be revealed with fire, and the fire will test the quality of each person’s work. 14 If what has been built survives, the builder will receive a reward. 15
If it is burned up, the builder will suffer loss but yet will be saved—even though only as one escaping through the flames.
So YES! the carnal believer remains a child of God.
So YES! the carnal believer will suffer loss in eternity, as declared by Scripture.
And YES! the carnal believer will face discipline here on earth.
As to what heaven will be like for him... It will be just as amazing as heaven can be. He will not, however, receive rewards. As to what losses the unrepentent *believer* will face here on earth, I will paste this list for you:
The Present Consequences of Sin or Disobedience
While the following is not exhaustive, it demonstrates that sin in the life of a believer is not a small issue.
(1) Loss of Fellowship With the Lord. Known sin in one’s life causes a loss of intimate fellowship with the Lord with the consequent loss of His joy and peace ().
(2) Divine Discipline From the Lord Here in Time. We should not think of discipline as punishment. Discipline from God is the gracious work of a Father to train and develop His children. Sometimes this comes in the form of various kinds of testing, trials, failure, and predicaments which He uses to correct us, to train us, and, if we have been going our own stubborn way, to increase our misery. The goal, however, is always to bring us back to Him (). If the believer remains unrepentant, this can lead to the sin unto death as with Ananias and Sapphira (), and some of the believers at Corinth who were failing to confess their sin and get right with the Lord (f; cf. also ).
(3) Loss of Power and Production. When we fail to deal with our sinful ways through honest confession, we grieve the Spirit’s person and quench His power in our lives. This means that rather than operating by faith in God’s provision, we end up operating in the energy of the flesh. We turn to our personal bag of tricks by which we seek to handle life (; ). This results in the works of the flesh and their awful and fruitless consequences (, 26). Without the abiding life, the life of faith and obedience to the Savior, we can do nothing ().
(4) Loss of Opportunities. When we are in charge of our lives rather than the Lord, we become insensitive to people and opportunities of ministry—we lack vision. Carnal believers have no vision other than their own personal agendas and selfish goals (cf. f).
(5) Loss of Desire and Motivation for Service. Carnal believers are occupied and controlled by their own self-centered desires (f). Perhaps this is a good place to discuss the concept of selfishness and rewards for some see an appeal to rewards as selfish and therefore carnal.
from: https://bible.org/article/doctrine-rewards-judgment-seat-bema-christ
The temporary, NON-eternal flames you (rightly) speak of, purifying the saved soul, are in some NON-eternal place or state. It is a place or state of purification. Which is the exact meaning of the term we use, "purgatory."