Responding to the OP's concern for his friend's view, which was:
"What seems insurmountable to me is that the new covenant is made in Christ’s blood. A person cannot be in it unless His blood be applied to them. And that blood is only applied to the elect at the moment of faith. So the paedobaptist view inevitably leaves you with unbelievers who are somehow related to Christ in some other way than a saving faith-union with him. I don’t find anywhere that admits of such a relationship."
The sentences, "And that blood is only applied to the elect at the moment of faith. So the paedobaptist view inevitably leaves you with unbelievers who are somehow related to Christ in some other way than a saving faith-union with him.", are significant. That is, are they true or not?
In a sense, it is true that the blood is only applied to the elect at the moment of faith, as before that moment I was a child of wrath: "Wherein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience: Among whom also we all had our conversation in times past in the lusts of our flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others." (Eph 2:2, 3).
But in another sense it was applied to me from eternity:
"According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love: Having predestinated us unto the adoption of children by Jesus Christ to himself, according to the good pleasure of his will, To the praise of the glory of his grace, wherein he hath made us accepted in the beloved." KJV (Eph 1:4,5,6)
That it was
to be applied to me from eternity and not
merely "at the moment of faith" is clear. So what about the statement in the OP of, "unbelievers who are somehow related to Christ in some other way than a saving faith-union with him."?
Unbelievers are
not related to Christ in the manner just stated. In the OT sign and seal of the covenant of grace, which was circumcision, all the children were not necessarily elect,
but for the sake of the elect children among them, all were circumcised. Exactly the same in the New Covenant period is the case: for the sake of the elect children among them, all are baptized. The others, the reprobate among them, were not in God’s covenant, despite appearances.
As regards the "covenantal Baptist perspective" on this, it is made clearer in Joel Beeke and Mark Jones’
A Puritan Theology, where they write
:
“The debate [between baptists and paedobaptists] focused on how the Abrahamic covenant relates to the new covenant. The question…is whether we may speak of the Abrahamic covenant (singular, so the Reformed) or Abrahamic covenants (plural, so the Baptists). The antipaedobaptists had to speak of two covenants made with Abraham: works and grace. By doing so they were able to argue that circumcision belonged to the Abrahamic covenant of works and not to the Abrahamic covenant of grace. Reformed paedobaptists would view this as forced exegesis that is wholly unpersuasive—particularly in light of Romans 4:11—and a major departure from classic Reformed covenant theology.” (A Puritan Theology: Doctrine for Life, Beeke and Jones, p 740-41)