What is the view of Reformed Protestantism on those who were 'numbered with' the apostles but did not, as did Paul, receive Peter's approbation?

Appendix 189 in The Companion Bible (Bullinger reprint of what I take to have been the 1885 edition) deals with Apostles, and confirms much of what has been claimed. Relevant parts state:

"In Matthew, Mark, Luke, and Acts (except 14:4,14) the term is used of the Twelve chosen and commissioned by the Lord (Matt. 11:1, Luke 6:13) during His earthly ministry. From this office Judas fell, his place being filled later by Matthias (Acts 1:26). In the Epistles and Revelation, the context shows where the Twelve are meant. The one occurrence in John's Gospel is in 13:16, where it is used in the general sense of one sent forth (on some special message or errand)...

Besides the Twelve there were others appointed by the Lord after His Ascension (Eph. 4:11 cp. 1 Cor. 12:28). Such were Paul and Barnabas, first called so in Acts 14:4,14; Andronicus and Junias (Rom. 16:7). Paul nineteen times calls himself an apostle, and argues his claim in 1 Cor. 9 and 2 Cor. 12. See also 1 Thess. 1:1; 2:6 where Paul associates Silvanus and Timothy with himself."

Then I looked at the notes for Rev. 21:14, "And the wall of the city [Heavenly Jerusalem] had twelve foundations, and in them the names of the twelve apostles of the Lamb." To my surprise, the notes said:

"apostles - The twelfth will be Matthias, not Judas. Twelve is the basic number of the measurements of the city."

It was the first sentence that surprised me, not the second one. Given what we know of the apostle Paul, chosen directly by the risen Christ, I would have thought Paul's name would be the twelfth one, not Matthias. But nobody is going to be able to know the twelfth name until they find themselves in this symbolic, spiritual city! What is obvious is that the whole of Revelation is full of numeric symbolism; heavenly Jerusalem represents "the bride, the Lamb's wife" (21:9) and the number twelve constantly recurs when describing the vision of the city: twelve gates at which were twelve angels "and names written thereon, which are [the names] of the twelve tribes of the children of Israel" (vs. 12), then the twelve foundations of the walls with the twelve names of the apostles of the Lamb, and twelve precious stones garnishing the foundations of the walls (vs. 19). Then the measurements are multiples of twelve.

Going back to the vision of the city, here is an interesting comment connecting the twelve gates that have the twelve names of the children on Israel written on them, with the twelve names of the apostles of the Lamb:

"Twelve is the number of the covenant, whether old or new. The twelve gates, twelve tribes, and twelve angels allude evidently to the old covenant people of God. But not after the flesh... they which are of the flesh, these are not the children of God: but the children of the promise are counted for the seed (Rom. 9:8). It is these, and these only, whom the twelve gates indicate... Throughout the old covenant... by such angels Christ indicated the promise of the gospel... Christ spake to his servants Abraham, Isaac, and Jacob by angels. Angel after angel appeared to the prophets, foretelling of the coming of Christ and of the evangel." (p 596)

So, the twelve names of the Apostles would speak directly of the new covenant. But because the twelve gates and the twelve angels speak of the old covenant, the connection is made - those who are spiritual 'children of Abraham' include many natural Jews and all spiritual Israelites. This gives the sum total of every single person who lived by faith and pleased God. All such will be part of this heavenly new Jerusalem, the Lamb's wife.

In a sense, that means that the question of who are 'numbered with the apostles' is comparatively irrelevant, for once we get to the book of the Revelation, it is the sum total of all who walked by faith who are represented by the twelve names of the twelve tribes of Israel, and the twelve names of the apostles of the Lamb. They will all form part of this heavenly Jerusalem, the Lamb's wife. I wonder if the unusual naming of the twelve tribes of Israel in Revelation 7:4-8 might hint at perhaps unusual naming of the twelve apostles of the Lamb in 21:14? But I for one am content to leave discovery of what that twelfth apostolic name is, to the marriage supper of the Lamb.

This falls short of your request to "know the mind of those who have led Reformed Protestantism as to the authoritative stance" but that's because I have chosen to answer in light of what the book of Revelation says. I hope someone more knowledgeable about Reformed / Puritan explanations of the various apostles can supply quotations to suit.

If Matthew Henry (1662-1714), a Protestant Nonconformist minister, is considered representative of the Reformed Protestant tradition, then the following extract may be useful. Regarding John’s vision of the New Jerusalem, Matthew Henry speaks of the wall for security, its purity and perfection, its size and capacity to house all the people of God, and then says this:

[5.] The foundation of the wall, for heaven is a city that hath her foundations (v. 19); the promise and power of God, and the purchase of Christ, are the strong foundations of the church's safety and happiness. The foundations are described by their number—twelve, alluding to the twelve apostles (v. 14), whose gospel doctrines are the foundations upon which the church is built, Christ himself being the chief corner-stone; and, as to the matter of these foundations, it was various and precious, set forth by twelve sorts of precious stones, denoting the variety and excellency of the doctrines of the gospel, or of the graces of the Holy Spirit, or the personal excellencies of the Lord Jesus Christ. https://www.christianity.com/bible/commentary.php?com=mh&b=66&c=21

Another Reformed Protestant view is to be found in the book, ‘The Revelation of Jesus Christ’ by John Metcalfe, under the section pertaining to the Seventh Opening. In Revelation 21:9-10 John is carried away in the spirit and sees the great city, holy Jerusalem, descending out of heaven. The vision is spiritual, not natural. The revelation of the holy city itself is to be envisaged metaphorically and interpreted spiritually (page 592).

What is being conveyed transcends all human thought, imagination, conception, idealism, comprehension or capacity. It is after this world has ceased, beyond time, past the existence of this age, and outside the limitations of the soul and body now existent (page 593).

With regard to the 12 foundations, after reminding us that 12 is the number of the covenant (whether old or new), Metcalf goes on to say:

This is used as a simile of the vital importance of the apostolic foundations of the holy city, New Jerusalem. These foundations are objective: they stand in the apostles’ doctrine... This foundation is the doctrine of Christ (page 598).

This is confirmed in Acts 2:42:

And they continued steadfastly in the apostles’ doctrine.

The wall is called Salvation, but the foundations rest on the New Testament doctrine of the apostles of the Lamb (page 599).

I can only conclude that the 12 foundations are not named after men, but are representative of the gospel, the doctrine of Christ, the salvation that comes through “the faith once delivered to the saints” (Jude verse 9).

Their names are not important, because the gospel points, not to men, but to Christ Jesus, the Lamb of God as revealed in the vision granted to John.