From a Roman Catholic point of view, what should a Protestant chaplain do for a dying Catholic?

I would suggest that the best course of action here is to contact the Chancery office of the Roman Catholic diocese in which you live or work, and pose the question to them. It may be that some form of extraordinary process can be worked out. I don't know how true this anecdote is, but I was told years ago by a Lutheran Pastor who had served in a Western state, where the only other parish in town was a Roman Catholic parish, and where that Parish was quite distant from the nearest Catholic Parish, that he had been irregularly authorized by the Bishop with authority over the Parish to provide certain short term services to the Parish during an interval of a month or two after the Priest in charge died unexpectedly, and a replacement Priest was not immediately available. In any case the Chancery office would have more authoritative information than we do here.

Some years ago, I befriended an inmate who was scheduled to be executed and was ask to be at his execution, in order to give him any support I possible could. I prayed the act of contrition with him and encouraged him to trust in Jesus' Merciful love for sinners.

In the absence of a priest, which was my case, I prayed for the inmate the best I could.

Many small Catholic Missals have a selection of prayers for the dying, usually just before or after the various prayers for the dead. One is quite free to follow the dictates and promptings of the Holy Spirit in the choice of prayers to use, but a Perfect Act of Contrition is most important, since this prayer directly prepares the soul for its voyage home: heaven. If you, as a Protestant chaplain are comfortable reciting Catholic prayers, in the presence of the dying, then I would recommend saying the following prayers.

If the Catholic knows how to make a perfect act of contrition, this would be the ideal prayer to begin with. If not, one can teach it to him or pray it with him. There are many version of this prayer and here is an example:

Forgive me my sins, O Lord, forgive me my sins; the sins of my youth, the sins of my age, the sins of my soul, the sins of my body; my idle sins, my serious voluntary sins; the sins I know, the sins I do not know; the sins I have concealed for so long, and which are now hidden from my memory.

I am truly sorry for every sin, mortal and venial, for all the sins of my childhood up to the present hour.

I know my sins have wounded Thy Tender Heart, O My Savior, let me be freed from the bonds of evil through the most bitter Passion of My Redeemer. Amen.

O My Jesus, forget and forgive what I have been. Amen.

One should encourage him to focus on Christ and His Passion:

Ways of Comforting the Dying and Encouraging Him to Focus on Christ and His Passion

Throughout the dying person's time in his sick bed, it is good to offer to him short acclamations that encourage the focus on Christ and the prayers of the Saints. Below are the ones listed in the 1958 St. Andrew Daily Missal. Periodically whisper these prayers into the dying person's ears:

Into Thy hands, Lord, I commend my spirit.

O Lord Jesus Christ, receive my spirit.

Holy Mary, pray for me.

Holy Mary, Mother of grace, Mother of mercy, do thou defend me from the enemy, and receive me at the hour of death.

It is recommended, too, to read the 18th and 19th chapters of the Gospel according to St. John to the dying, helping him to focus on Christ.

If your are still comfortable reciting more Catholic prayers you could say the prayers Commending the Soul to God.

Here is an article for Catholics (Eucharist Ministers) on how they should give the Sacraments for the Dying in the Absence of a Priest (Viaticum). Although only for Catholics, it does point out that one should at least have the dying person pray a Perfect Act of Contrition.

"Can. 916 Anyone who is conscious of grave sin may not celebrate Mass or receive the Body of the Lord without previously having been to sacramental confession, unless there is a grave reason and there is no opportunity to confess; in this case the person is to remember the obligation to make an act of perfect contrition, which includes the resolve to go to confession as soon as possible." Code of Canon Law, Sanctifying Office of the Church, Participation in the Blessed Eucharist.

(Being in danger of death should certainly constitute a "grave reason" and the "absence of a priest" would mean no opportunity to confess.)