What is the Reformed Protestant Interpretation of 1 Timothy 1:19 in relation to “eternal security”?

To be shipwrecked in regard to the faith is not necessarily to be damned. It will, certainly, make the person professing Christian faith become totally unsure of their salvation, but if they (like the prodigal son) repent and come back to the Father, their assurance of salvation will return. This is because one of the works of the Holy Spirit is to convict of sin; to reprove those whom God chastises. He withdraws from those who grieve him so that they feel his absence. The darkness of having entered into sin fills them with dread once they experience God's discipline which awakens them to this. But that is only true of those who have been born again from above, of the Spirit.

Another group who never were born spiritually were spoken of by the Lord, here:

"Not every one that saith unto me 'Lord, Lord', shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven. Many will say to me in that day [of judgment] 'Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?' And then I will profess unto them, 'I never knew you; depart from me, ye that work iniquity'." (Matthew 7:21-23)

Notice how they never were known by Christ? Oh, he knew all about them and their miracles and their works, all claimed to be done in Christ's name. But he never knew them in the sense of acknowledging them as his own, as being children of the kingdom. They thought they were - they professed to be Christians - they impressed many people with their miracles and their works - but they never did belong to Christ for they never had saving faith in him.

Once we bear in mind those two groups of people, then we can see how the statement in 1 Timothy 1:19 applies to the question of whether there is truth in the Reformed doctrine of the eternal security of the believer.

You correctly alluded to some commentators who say it speaks of remedial correction. The 1987 edition of the N.I.V. Study Bible notes that 1 Tim.1:3-11 speaks of heretics creeping in as wolves, but speaks differently about Hymenaeus and Alexander who are handed over for church discipline. Paul had excluded them from the congregation.

"Out in the world, away from the fellowship and care of the church, they would be 'taught' (the word means basically 'to discipline') not to blaspheme. The purpose of such drastic action was more remedial than punitive." (N.I.V. Study Bible p.1799-00)

In the late 1700s or early 1800s, a Reformed preacher wrote a meditation on the work of the Holy Spirit, and how he is "the seal of God" within the believer. Regarding spiritual gifts, such as working miracles, he wrote:

"Spiritual gifts... are not the things that accompany salvation; men may have all these, and yet be nothing - 1 Cor.13:1,2 [then he quotes Mat.7:21-23].

Balaam was a worker of iniquity when he prophesied, and so was Judas when he preached and wrought miracles; the love of money, the root of all evil, was in the heart of them both; nor did they ever love God, or seek his glory; they sought their own glory, and therefore they were not true men, nor was there any righteousness in them." Divine Meditations, pp178-9, William Huntington

However, he did not put the likes of Hymenaeus and Alexander into that category. He wrote of how the apostle Paul was "all things to all men, that I might by all means save some":

"To the Thessalonians Paul was a nurse, to Timothy an affectionate father... and to Alexander and Elymas the sorcerer he was a lion." (Ibid. p231)

On the question of whether such men were "sealed" unto God, he elaborates:

"Hymeneus and Philetus, concerning the truth have erred, saying that the resurrection is past already; and overthrow the faith of some. Nevertheless the foundation of God standeth sure; having this seal, 'The Lord knoweth them that are his'. II Tim.2:18, 19

...Hence the apostle concludes that, though some, who confess the name of Christ, depart from the faith themselves, and overthrow the faith of others, yet the true Messiah, who is the omniscient God, and who, as man, is sealed with the Holy Ghost, must know who his own elect are; he must know whose sins he bore, for whom he died, and who are his own purchased possession; and surely he will never lose what his Father has give him, nor suffer any of those to be plucked out of his hand who are made his charge. The good Shepherd may suffer Satan to steal away the wolves, but not the sheep; for the Lord, but none else, knoweth them that are his." (Ibid. pp184-5)

Yes, some may depart from the faith and some may even stumble the faith of others, but those who do that yet are known to the Lord as sealed unto him, will not lose salvation; they may repent and return to the father; those whom they stumbled may regain their footing. Huntington does not say those particular men were 'wolves', just that the Lord will not allow any wolves in the flock to steal away the sheep - i.e. those who are sealed by the Holy Spirit (John 6:27).

A modern Reformed theology book has this to say about 1 Timothy 1:19, making a fine point about the phrase "the faith" as opposed to a personal act of faith (belief):

"In other cases we find references to "the faith" - that is, the doctrinal content (Ac 6:7; 1 Ti 1:19; 3:9; 5:8; 6:12; Jude 3). Therefore the distinction often made in theology between the faith that is believed (fides quae creditur) and faith as the personal act of believing (fides qua creditur) seems well founded. This means that the personal act of faith has an object (Christ as he is clothed in the gospel), a content (the doctrine concerning Christ and his gospel), and a subject (the believing sinner)." Pilgrim Theology, p.266, Michael Horton

Check out all those references he gives. God knows those who deny "the faith" that is believed. God knows those who made a personal act of faith, believing in Christ. But we cannot judge because we do not know what only God knows about their standing before him. We know that Jesus warned there would be some who, to all appearances, were truly believers, but who he never did 'own' as belonging to him. And we also know that some believers fell away for a season, only to repent and return to their heavenly Father.