You said: "When we bear this mark, God the Father recognizes us as His sons." But if that were true, it would mean God recognizes as his sons the baptized souls in hell, which is absurd. It is the justified souls – i.e., the souls that participate in the divine life of God (sanctifying grace) – who are the adopted children of God, not those who possess the baptismal character - many of whom are cut off from God for all eternity and suffering the eternal pains of hell..
You miss the distinction between in potency and in act. And, yes, in fact, some saints talk about the horror of someone marked with the character of Baptism being in hell, bearing that mark on his soul. You'll also find priests in hell who are ontologically configured to the likeness of Christ, forever,
in persona Christi.
You mentioned that the character gives us a participation in Christ. How? In the priesthood, the person of the priest becomes ontologically conformed to the person of Christ. Same, to a lesser extent, with the Baptismal character.