According to a ‘forgiveness in Christ alone’ framework how did ‘sin offerings’ make atonement for sin under the Mosaic Law?

Briefly, this is exactly the problem that dispensationalism was trying to address. Scofield, Darby, et. al hypothesized that at different times, there were different ways to secure Gods favor. Key to this concept were the ideas of covenant and progressive revelation.

Progressive Revelation, in a nutshell, says that the Bible should be considered chronologically. If it "hasn't happened yet" in Scripture, then God hasn't yet revealed that as a condition or requirement. In other words, covenants must be explicitly borne out. Since Grace or even sole fide are not explicitly mentioned as being covenantial, an alternative dispensation (namely the sacrificial system) was required.

Depending on the scheme proposed by the dispensationalist, there were different covenants in place. Before the flood, it was to have fellowship with God. During the days of Israel it was through sacrifice. During the church age, it is through grace. During the tribulation it will be through avoiding the mark. The point is that dispensationalism teaches that history provides different mechanisms at different times.

Dispensationalists look to Hebrews as evidence of this system. Hebrews claims that Jesus was the fulfillment of the sacrificial system. Look, for example, at Hebrews 10:1

The law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship.

In other words, the covenant relies on ongoing sacrifice.

In being the sacrifice on the cross, Jesus, the High Priest overturns the sacrificial system. It says in v 12 - 14:

But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God, 13 and since that time he waits for his enemies to be made his footstool. 14 For by one sacrifice he has made perfect forever those who are being made holy.

With this sacrifice, Jesus is able to overturn the old covenant. As it says in verses 15ff:

15 The Holy Spirit also testifies to us about this. First he says: 16 “This is the covenant I will make with them after that time, says the Lord. I will put my laws in their hearts, and I will write them on their minds.”

In short, under this system, God changes the rules. Pure and simple.

I think the only thing we can say for certain is that if the offerer presenting a sacrifice to atone for sins, did not do it with faith in Messiah (who actually providing that atonement) then no such atonement was obtained, except possibly an external kind of 'accepting the worshiper into the temple practices. '. In other words faith in Christ alone is the only way to obtain atonement but the external atonement of being accepted into the external form of worship, symbolizing true atonement, could be obtained by an animal. If we at any time imagine that real atonement for sins could be obtained in any other way than faith in Messiah, Christ died in vain. If atonement and favor of God could have been obtained without a direct link to Christ's death, then Christ did not need to die as that method could have obtained salvation without any further help by a God-Man.

Therefore these different ways of obtaining God's favor must have been expressions of faith just as the Lords supper is for us. Atonement and God's favor is not at all obtained by them, but faith in Messiah is expressed by them and that has always been how man found favor with God and no other way is possible and capable of atoning for sins. Sin deserves infinite punishment for offending the glory of an infinite God, making sin an infinite crime. Only the God-Man could endure such a punishment. In fact the smallest sin could never be even partially atoned for by all the goats, lambs and bulls ever born from Adam to now. These were symbols not realities as Hebrews argues and as the Lords supper proclaims.

Protestants have two sacraments, baptism and the breaking of bread, but they had hundreds of sacraments under a symbolic and externalization of faith representing a reminder of sins and a promise of better things.

The law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship. Otherwise, would they not have stopped being offered? For the worshipers would have been cleansed once for all, and would no longer have felt guilty for their sins. But those sacrifices are an annual reminder of sins. (Hebrews 10:1-3, NIV)

However due to the waekness of that system by its use of animals to pay for human sins the worshipper could never have full confidence. They knew the object was not appropriate for the need, like to not answer like. Furthermore the clearness of the meaning of Messiah was still to vague in their minds. Therefore the sacraments were not fully 'able to clear the conscience of the worshiper.' This must have been part of the burden that Jesus wanted to alleviate when he said 'come unto me all whomare heavy laden and I will give you rest'.

Here we see the New Testament light on those ancient practices:

When everything had been arranged like this, the priests entered regularly into the outer room to carry on their ministry. But only the high priest entered the inner room, and that only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance. The Holy Spirit was showing by this that the way into the Most Holy Place had not yet been disclosed as long as the first tabernacle was still functioning. This is an illustration for the present time, indicating that the gifts and sacrifices being offered were not able to clear the conscience of the worshiper. They are only a matter of food and drink and various ceremonial washings —external regulations applying until the time of the new order. (Hebrews 9:6-10, NIV)

How did the old ceremonial sacrifices made atonement?

They didn't. Old Testament sacrifice was only able to atone for someone who followed the Mosaic law perfectly... a feat which is beyond any of us, especially as most of us are not Jews, and even if we were the Altar and priests are gone. The whole point of Jesus' sacrifice is that the former sacrificial system was inadequate. Sin offerings succeeded only until "just the right time", when Christ made one sacrifice for all.