How does Catholicism biblically defend purgatory in a way that upholds the sufficiency of Christ's work?

My question is not about the prayers or the process of purification in purgatory. My question is about the notion of there being some aspect of Divine Justice that was not satisfied on the sinners behalf by Jesus Christ!

St. Paul can give us some relevant insight here, since he speaks directly to this issue (by direct implication about Christ's passion).

He wrote:

Who now rejoice in my sufferings for you, and fill up in my flesh those things that are wanting with regard to the sufferings of Christ, for the sake of his body, which is the church (Colossians 1:24)

What did St. Paul mean when he said the passion of Christ was "wanting" something in addition? How can we - Paul - add to Christ's work, in order to accomplish something for believers? It should be obvious that St. Paul means Christ left some work to us willingly. In which case the 'insufficiency' is not from want of power or efficacy, but from will - Christ wants us to participate.

Jesus expresses as much in the following words:

Then Jesus said to his disciples: If any man will come after me, let him deny himself, and take up his cross, and follow me. (Matthew 16:24)

This isn't a suggestion, but a command. In what sense do Christians need or want to humble and punish themselves by self-denial? Unless Christ invited us to partake in Him and his anti-sin offering of sacrificing what He was entitled to, in a similar way to how we partake in Adam by our sin, having what we are not:

...by one man sin entered into this world, and by sin death; and so death passed upon all men, on account of whom all have sinned. (Romans 5:12)

Another point needs to be made clear: all good things come from God. Including being made pure through suffering, in purgatory, or on earth. When God says, "I will try/test" He is the agent of the perfection of those who persevere and endure and are found better on the other side than beforehand. Not the people who endure - who can endure only by His help and grace.

This is the new covenant: "Heb 8:12 For I will be merciful toward their iniquities, and I will remember their sins no more.

What God means by "I will remember their sins no more" is 'I will forgive them their sins and afterwards never accuse them of those sins.' Those who are in purgatory are forgiven all of their sins already, or they wouldn't be there.

The punishment due to offending God, and the guilt itself of committing the offense, are separate. We read this in the story of David and his sin, and his punishment - although forgiven the guilt - of that sin.

And David said to Nathan: I have sinned against the Lord. And Nathan said to David: The Lord also hath taken away thy sin: thou shalt not die. Nevertheless, because thou hast given occasion to the enemies of the Lord to blaspheme, for this thing, the child that is born to thee, shall surely die. (2 Samuel 12:13-14)

David gave up something by his sin which, even though he was forgiven for doing evil, God exacted for himself the justice of revenge on David for insulting His name. This is just.

Even though a Christian is forgiven for sinning, God has the right to demand that we have love for him and his forgiveness, which is just to do when forgiven freely and without need (just requirement), by willing to 'undo' the sins, or the equivalent thereof (i.e. suffering).

Even in this regard, Christ's sacrifice offers freedom from even this, since it is all-abundant. And Christians can benefit from the Eucharist, removing all their temporal punishment by a perfect devotion thereto, and an amendment of life.

Because as God said,

Go then and learn what this meaneth, I will have mercy and not sacrifice. For I am not come to call the just, but sinners. (Matthew 9:13)

What God meant wasn't 'I made a mistake and regret instituting sacrifice' nor that simply shrugging and regretting what you did is sufficient to blot out your sins on its own, but that He would rather we use the grace he provides to do good and avoid the need for sacrifice in the first place - "Go and sin no more." As if somewhere were to say, 'I would rather have no threats, and no second amendment.' Jesus is saying, 'I wouldn't be here among the worst of society, if everyone left sin and actually followed God!' i.e. including them - a strike at the conniving Pharisees.

St. Paul wrote:

For we must all be manifested before the judgement seat of Christ, that every one may receive for things done in the body, whether it be good or evil. (2 Corinthians 5:10)

And in his first epistle to Corinth:

If any man's work burn, he shall suffer loss; but he himself shall be saved, yet so as by fire. (1 Corinthians 3:15)

Those whose works are imperfect, not being built upon Christ, are burned up in God's trying justice, and such a person is saved indeed - yet so as by fire. Not literal fire, but as if he was burned to the last remaining good deeds which constitute his person. He brings with him only what he has left. And if sanctification always involves resistance - suffering - on earth, it will certainly do so - God's justice remaining the same, and demanding the same from those on earth, and not on earth - in the world to come. For the process of removing dross and leaving something pleasing to God is nothing other than the definition of sanctification.

When forgiven our sins, we must then at that point recognize the injustice of our ways, and repair that voluntarily, by God's help of course. Committing the opposite of wilful sin, by wilfully "deny"ing himself instead, as Jesus taught.

Another way of looking at it is that God forgives us, we recognize the injustice of such, and yet, now free from guilt by his mercy, we refuse to want to repay him somehow - such a sentiment is equivalent to not being thankful for forgiveness, which is itself another sin. And only sins which are repented of/turned away from are forgiven.

True, God is not bound to forgive, but those forgiven are bound to be thankful. And who, who is truly thankful for being forgiven, refuses to do anti-sin in place of the things in which they offended God?

The Catholic Church teaches the doctrine of Purgatory while also teaching the sufficient and superabundant merits of the Passion of Christ.

The first component to understanding this is the dogma that not all temporal punishment is remitted by God with the guilt of sin and the eternal punishment. This is based upon a number of examples in Scripture where despite the guilty party being forgiven, they receive a temporal punishment for their sin. For example:

And David said to Nathan: I have sinned against the Lord. And Nathan said to David: The Lord also hath taken away thy sin: thou shalt not die. Nevertheless, because thou hast given occasion to the enemies of the Lord to blaspheme, for this thing, the child that is born to thee, shall surely die. (2 Samuel 12:13-14)

The Church has connected this to Christ's admonition to take up your cross (do penance)

And he that taketh not up his cross, and followeth me, is not worthy of me. (Matthew 10:38)

Christ's atonement is fully adequate and superabundant:

For you are bought with a great price. ( 1 Corinthians 6:20)

Knowing that you were not redeemed with corruptible things as gold or silver, from your vain conversation of the tradition of your fathers: But with the precious blood of Christ, as of a lamb unspotted and undefiled, Foreknown indeed before the foundation of the world, but manifested in the last times for you, Who through him are faithful in God, who raised him up from the dead, and hath given him glory, that your faith and hope might be in God. (1 Peter 1:18-21)

Now the law entered in, that sin might abound. And where sin abounded, grace did more abound. (Romans 5:20)

Thus, the Church teaches that Christ's death merits sufficient grace to cover every sin, yet Divine Justice requires temporal punishment to make amends for the harm due to our sins. Our enduring temporal punishment has value because of the merits of Christ.

Finally, Purgatory as a necessity derives from Scripture:

And so betaking themselves to prayers, they besought him, that the sin which had been committed might be forgotten. But the most valiant Judas exhorted the people to keep themselves from sin, forasmuch as they saw before their eyes what had happened, because of the sins of those that were slain. And making a gathering, he sent twelve thousand drachms of silver to Jerusalem for sacrifice to be offered for the sins of the dead, thinking well and religiously concerning the resurrection, (For if he had not hoped that they that were slain should rise again, it would have seemed superfluous and vain to pray for the dead,) And because he considered that they who had fallen asleep with godliness, had great grace laid up for them. It is therefore a holy and wholesome thought to pray for the dead, that they may be loosed from sins. (2 Machabees 12:42-46)

Emphasis mine. If they were thought to be damned, no prayer could help. If they were thought to be saved, no prayer would be necessary.

2 Machabees is the most explicit as far as describing a practice with its theological implication, but there are New Testament passages which also have implications of Purgatory.

And whosoever shall speak a word against the Son of man, it shall be forgiven him: but he that shall speak against the Holy Ghost, it shall not be forgiven him, neither in this world, nor in the world to come. (Matthew 12:32; emphasis mine).

The "in the world to come" was interpreted by both Pope St. Gregory the Great and St. Augustine as referring to the possibility of some sins being remitted after death, hence a middle state of Purgatory.

Similarly, in 1 Corinthians:

According to the grace of God that is given to me, as a wise architect, I have laid the foundation; and another buildeth thereon. But let every man take heed how he buildeth thereupon. For other foundation no man can lay, but that which is laid; which is Christ Jesus. Now if any man build upon this foundation, gold, silver, precious stones, wood, hay, stubble: Every man's work shall be manifest; for the day of the Lord shall declare it, because it shall be revealed in fire; and the fire shall try every man's work, of what sort it is. If any man's work abide, which he hath built thereupon, he shall receive a reward. If any man's work burn, he shall suffer loss; but he himself shall be saved, yet so as by fire. (1 Corinthians 3:10-15)

Here there is a distinction between the man whose works abide (who shall receive a reward) and the man whose works burn (who shall be saved "by fire"). The latter case is understood as referring to purgation of the effects of their sins.

Be at agreement with thy adversary betimes, whilst thou art in the way with him: lest perhaps the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison. Amen I say to thee, thou shalt not go out from thence till thou repay the last farthing. (Matthew 5:25-26)

"repay the last farthing" was understood as referring to smaller offenses and the context suggests a time-limited rather than eternal penalty.