Covenant Theology is the framework of understanding Scripture through the lens of covenants. The Redemption Covenant being the primary oldest one and fulfilled in Christ Jesus. The other two are works and grace.
So, with that in mind, we know the reality that Christ purchased His people with His blood.
In whom we have redemption through his blood, the forgiveness of sins, according to the riches of his grace;
Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which he hath purchased with his own blood.
In turn, we find one shadow of this in the Abraham and Isaac and ram substitute sacrifice.
But to the OP, we find the primary purchase to which Moses mentions in the form of the Passover lamb. The blood of the lamb purchased the redemption of the people who believed.
In this sense, the nine other plagues of Egypt would also serve as lessons to some to believe God when He spoke about the 10th plague (the death of the firstborn, except for the blood of the Passover lamb). That is the purchase price.
And the blood shall be to you for a token upon the houses where ye are: and when I see the blood, I will pass over you, and the plague shall not be upon you to destroy you, when I smite the land of Egypt.
In the Song of Moses, Moses is speaking about the crossing through the sea, but as a confirmation of the aforementioned purchase to actually arrive there. First there's the purchase and then the deliverance.
Fear and dread shall fall upon them; by the greatness of thine arm they shall be as still as a stone; till thy people pass over, O LORD, till the people pass over, which thou hast purchased.
Those purchased via the lamb's blood passed through the sea, while the chasers, the unredeemed, the unpurchased were swallowed by the earth.
Powerful imagery as a foreshadowing of Christ our Passover Lamb.
Forasmuch as ye know that ye were not redeemed with corruptible things, as silver and gold, from your vain conversation received by tradition from your fathers; But with the precious blood of Christ, as of a lamb without blemish and without spot:
1 Pet 1:18-19