Was the Bundesbrief a forgery?

We cannot say for sure.

It would seem a stretch to claim that the Bundesbrief "is a forgery", from "the middle of the 18th century". (Christoph Pfister: "Die alten Eidgenossen: Die Entstehung der Schwyzer Eidgenossenschaft im Lichte der Geschichtskritik und die Rolle Berns", BoD, 2019. p84)

Jonathan Steinberg upholds his position from 1976 in later editions or when reviewing other works, like here in 2012.

But the transmission of the document is indeed very problematic, if not as problematic as its interpretation: A text of allegedly paramount importance, only 'discovered' in 1724 (or 1760)? The Nidwalden 'copy' (translation) being dated to 1400, but also discovered only in 1758.

Source criticism of this document is far from finished when listing the associated problems:

A physical format described as uncommon, a scribe and hand otherwise unknown from the region and time, language and details not really matching the surrounding time and location, all being a much better fit for a later date than 1291.

A diplomatic source criticism listing just a few of the problems and trying a limited reconstruction of the stages of creation is — Pascal Ladner: "Urkundenkritische Bemerkungen zum Bundesbrief von 1291", Mitteilungen des historischen Vereins des Kantons Schwyz, 83, 1991.

While wholesale forgery seems unlikely, just taking everything for granted isn't warranted either.

The biggest problem is how a supposedly 'hard science' approach seems to fog and sometimes suffocate the historical analysis.

Radiocarbon dating is not a perfect method. It depends on calibrations, models, mustn't be contaminated in any way. And above all, it can only give a hint based on a lot of presumptions. And the results operate with numbers that create the illusion of precision, while they only display probability clouds.

In this case, the actual dating results may be off for a variety of reasons. But if we assume the numbers arrived at are just that what we would see if the test would have been replicated with updated calibration curves: the exact same numbers. These tell us much less than we would like or even need to call the 'matter closed'.

The C14 test was done just once, on a small strip of parchmemt. The result being

85% probability 1252–1312
15% probability 1352–1385

Add to that the general error probability of these tests that with a probability of 5% the real age of the sampled material is out side of that window, either older than 1252 or younger than 1385!

The most important objection here under the assumption of near perfect methodology for the dating: the methodology for sampling and the underlying assumptions. A small strip of parchment was dated, but not the document. The document is parchment, plus writing, plus seals. Re-use, re-cycling or palimpsest are all very common phenomena for that period and completely uncontrolled for in this analysis. While the piece of parchment may very well be of the 'desired' and 'measured' age: if it simply lay in pile for while at a monastic warehouse, or a was clipped from a bigger and older document with 'ample whitespace', or… then we know the age of the material, but not the document or its content.

This fundamental critical assessment is a necessary addition to the radiocarbon results. In this case assembled by 'the historical component' of the radiocarbon analysis team:

For me, after all these considerations, there is no longer any doubt that the forgery issue must also be seriously addressed in the case of the Bundesbrief. Personally, I consider a "broken" development of the Federal Letter to be very possible and in any case no less likely than a development at the beginning of August 1291.
— Roger Sablonier: "Der Bundesbrief von 1291 : eine Fälschung? : Perspektiven einer ungewohnten Diskussion", Mitteilungen des historischen Vereins des Kantons Schwyz, 85, 1993. DOI (Still listing almost all relevant points, required reading for this answer.)

Take note that of those countless forgeries Steinberg may allude to a copy of this Bundesbrief in Zürich was once believed to be from 1251, and turned out to be a simple 'change one letter with a scraper' forgery that was dismantled in 1878.
_{— Review of: Brandstetter in: Der Geschichtsfreund. Mittheilungen des historischen Vereins der fünf Orte Luzern, (Uri, Schwyz, Unterwalden und Zug. 32. und 33. Band. Mit 2 resp. 5 artist. Tafeln. Einsiedeln, C. und N. Benziger. 1877 und 1878. (PDF)}

An accessible discussion of some problems in — Lucien Rahm: "Urserntal statt Unterwalden? Bundesbrief gibt Rätsel auf. Experten sind sich uneinig, wie das Original des Bundesbriefs übersetzt werden muss. Eine Auslegeordnung", Luzerner Zeitung, 26.08.2019. — Roger Sablonier: "Auch die Urschweiz gab es nicht", Neue Zürcher Zeitung, 27.07.2008.

In summary, we need to look at what we mean with 'forgery': a complete invention seems in this case implausible. A rewriting of an older accord, which may even have been made only orally, seems quite likely. The seals, locations mentioned and especially the date remain quite questionable. A later date for a full forgery is not impossible, but more likely is that mainly the given date of 1291 is not really accurate and 1309 historically 'a better fit'. To reiterate '1291' could be true, August 1st is a later inferring invention.

Regarding the head scratcher quote from the question:

The famous Bundesbrief of 1291 is one of many ‘Fälschungen’ – forgeries which carbon-dating has shown to be from a later period.

In light of the above I suggest to read that as:

The famous Bundesbrief of 1291 may be one of the many ‘Fälschungen’ – one of countless forgeries from that time for which carbon-dating has shown for many to be from a later period. But while many of those forgeries can be convincingly proven to have the wrong date attributed to them, the Bundesbrief itself has not been convincingly re-dated by radiocarbon dating alone.

or

The famous Bundesbrief of 1291 is one of many ‘Fälschungen’. And such forgeries have been regularly shown to be from a later period, via carbon-dating.

The original quote may be unlucky in its formulation, outdated, inviting to mislead, or simply untrue. The reformulated versions on offer here are not wrong. But Steinberg himself seems to have dialed back his assertion recently, concentrating on essentials:

… the so-called Bundesbrief, whatever its date, whatever its provenance, had one unique feature.
_{— Jonathan Steinberg: "Why Switzerland Is a “Special Case” and Cannot Be a Model for Other States “Why Switzerland?” — Switzerland through foreign eyes", Avenir Suisse, 22.07.2014.}

Despite the carbon dating results, there remains a degree of uncertainty as to whether or not the Bundesbrief (Federal Charter) of 1291 is a medieval forgery and, if it is, to what extent it has been forged. It seems that the most accepted (but certainly not definitive) answer is that it is a backdated document from the early 14th century with some additions. In other words, the date of the document may be forged, but not the core 'spirit' of it (despite the additions and possible amendments). This would not be inconsistent with the carbon-dating.

Looking at other sources on the Bundesbrief of 1291, Jonathan Steinberg appears to have omitted to update his book (the 2016 edition is the third - the 1st was in 1976) or else is using 'forgery' without specifying exactly what he means by it. According to one of your sources, (Winkler, 2014), and Clive H. Church's A Concise History of Switzerland (2013), the document may have been written in the early 14th century but was possibly based on a 1291 document (presumably now lost).

This is what Church has to say:

Purportedly drafted and sealed in early August 1291, the actual document displays numerous peculiarities, and may well have been written down in 1309 or even later;

Winkler goes into more detail, explaining

As Professor Sablonier observes, "Every Medievalist knows that hundreds of surviving sources from the Middle Ages (especially those that grant privileges and establish fundamental relationships) have been forged or contain inauthentic parts that have been intermingled [with accurate materials] or been placed later [in the source] ."24 While Dr. Sablonier never states that the Federal Charter is a fake, he suggests that its contents may have been created between 1320 and 1330, and he also indicates that the internal evidence in it makes 1309 as the most likely date for its composition. If so, the manuscript was probably given the date of 1291 to make it appear more noteworthy because it was older.

Winkler, though, does not rule out the possibility that it is a forgery as there are inconsistencies, but he further states that:

...the parchment upon which the Federal Charter was written is clearly of late medieval origin, but the date of this material may differ from that of the source's content. The parchment could have been clipped from another document written at the appropriate time, and the text could have been added later. The fact that the manuscript has relatively narrow margins, especially on the right and left sides, could be evidence that the material was cut from another source and was the wrong size for its intended purpose. However, the strongest indication that the content of the Federal Charter dates no later than the fourteenth century is the existence of the Nidwalden Copy from roughly 1400.

The authors of 2016 article, On the Institutional Roots of Swiss Democracy are among the academics who (tentatively at least) accept the likelihood of an early 14th century date. There has also been some comment on the actual importance of the Federal Charter (whether it dates to 1291 or the early 14th century) by Roger Sablonier in Chapter 25 of the Cambridge Medieval History (vol.7), and this is echoed in this Federal Council (government portal) article, The Federal Charter of 1291:

For centuries, this alliance of the valleys of Central Switzerland from 1291 received practically no mention, the document itself only being rediscovered in 1758 in the Schwyz archives. In the constitutional tradition of the Old Confederacy prior to 1798, the alliance of 1291, in contrast to the Federal Charter of 1315 (Pact of Brunnen), played no role. Nor is there any indication that it served as a model for subsequent federal alliances. Unanswerable questions relating to its historical provenance and interpretation also suggest that the contemporary importance of this document in more recent times has been considerably overestimated.

The article adds, though, that

An appropriate assessment of the historical circumstances does not detract from the enormous cultural value of this prominent document, which is still referred to regularly in present day political debates.