Trvia: the Challoner actually gets its more poetic style from the KJV (as Bishop Challoner was himself an Anglican convert). So the process goes full circle: the translators of the KJV had themselves borrowed from the original Rheims New Testament of 1582, and now Challoner is borrowing from the KJV. See for example Ephesians 3:6-12:
Rheims: † The Gentils to be coheires & concorporat and comparticipant of his promis in Christ IESVS by the Gospel: † vvhereof I am made a minister according to the gift of the grace of God, which is giuen me according to the operation of his povver. † To me the least of al the sainctes is giuen this grace, among the Gentils to euangelize the vnsearcheable riches of Christ, † and to illuminate al men vvhat is the dispensatiō of the sacrament hidden from vvorldes in God, vvho created al things: † that the manifold vvisedom of God, may be notified to the Princes and Potestats in the celestials by the Church, † according to the prefinitiō of worldes, vvhich he made in Christ IESVS our Lord. † In vvhom vve haue affiance and accesse in confidence, by the faith of him.
KJV (1611): That the Gentiles should be fellow heires, and of the same body, and partakers of his promise in Christ, by the Gospel: whereof I was made a Minister, according to the gift of the grace of God giuen vnto mee, by the effectuall working of his power. Unto mee, who am lesse then the least of all Saints, is this grace giuen, that I should preach among the Gentiles the vnsearchable riches of Christ, and to make all men see, what is the fellowship of the mysterie, which from the beginning of the world, hath bene hid in God, who created all things by Iesus Christ: to the intent that now vnto the principalities and powers in heauenly places, might be knowen by the church, the manifold wisedome of God, according to the eternall purpose which he purposed in Christ Iesus our Lord: in whom we haue boldnesse and accesse, with confidence, by the faith of him.
Challoner: That the Gentiles should be fellow heirs and of the same body: and copartners of his promise in Christ Jesus, by the gospel, of which I am made a minister, according to the gift of the grace of God, which is given to me according to the operation of his power. To me, the least of all the saints, is given this grace, to preach among the Gentiles the unsearchable riches of Christ: and to enlighten all men, that they may see what is the dispensation of the mystery which hath been hidden from eternity in God who created all things: that the manifold wisdom of God may be made known to the principalities and powers in heavenly places through the church, according to the eternal purpose which he made in Christ Jesus our Lord: in whom we have boldness and access with confidence by the faith of him.