According to Reformed Covenant theology, how were the people of Israel "purchased" per Exodus 15:16? In the triumphal song of Moses, it is mentioned in 15:15 that the people were purchased.


  Exodus 15:16 Terror and dread fall upon them;         because of the
  greatness of your arm, they are still as a stone,     till your people, O
  LORD, pass by,        till the people pass by whom you have purchased.
  (ESV)


"Purchase" usually implies an exchange of one valuable thing for another valuable thing. Purchasing language becomes important in the New Testament for understanding the sense that the Lord Jesus purchased his people with his own blood. In the Exodus, there is no clear exchange taking place. The Exodus appears mainly to be a triumph of the Lord against his enemies. Yet Moses sings of a purchase.

How does Reformed covenant theology understand the purchase here described? Is this a foreshadowing of the ultimate purchase of God's people with the blood of Christ, or is there a more immediate sense that a purchase has taken place?
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;
  Eph 1:7
  
  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.
  Acts 20:28


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.
  Exo 12:13


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.
  Exo 15:16


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