I love chasing down references like this, so let’s do some digging.

The entry you quote is written by Allan Chapman (shoutout to my homie), but as you also noted, he doesn’t mention anything else about the “legend” in the text. In the Bibliography for the entry he cites three works. The first one is an English language translation of the Epistola by Peregrinus, which edition didn’t include any substantial additional explanatory text. The second one is an entry from the Dictionary of Scientific Biography. The last one is a text in German from 1898, which I’ve been unable to check due to the language barrier.

The Dictionary of Scientific Biography is available to borrow for free on archive.org. The entry for Peter Peregrinus is lengthy, but it includes the following exciting lines:

“Although there is evidence that Peregrinus was of noble birth,(6) the suggestion that he was a theologian is unconvincing(7) and the assertion that he was a Franciscan is baseless.(8)”

Reference 7 leads to “F. Picavet, Essais sur l’histoire generale et compare des theologies et des philosophies medievales, 240-242, 252”. Just like my German, my French is atrocious, so I’ll let you try and read this.

Reference 8 leads to “Stewart Easton, Roger Bacon and His Search for a Universal Science (Oxford, 1952), 120-121.” This book is available for free through archive.org, and it discusses in more length the “legend” or debate. On page 120 Easton mentions that another historian, Edward Hutton, “comes to the interesting conclusion that Peter de Maricourt and Bacon’s friendship for him [i.e. for Bacon] were deciding factors [for joining the Franciscan Order]. For Peter, according to Hutton, was a Franciscan.” If we check Hutton’s book, his reasoning is that Bacon wrote highly of Peregrinus, which means that they were close. In addition, Hutton is so certain that Peregrinus was a Franciscan that he states:

“As it happened, this man was a Franciscan.”

In Hutton's characterisation of the circumstances, Peregrinus was like a saviour who lifted Bacon into the Franciscan Order. Of course, he does not provide any references or evidence to his claims. My interpretation of Hutton’s argument is that it rests on both Peregrinus and Bacon being located in Paris for a period of time, as well as on Bacon’s praise of Peregrinus. However, Easton argues that (besides Peregrinus not being a Franciscan) it was also unlikely that Peregrinus and Bacon had known each other before Bacon entered the Franciscan Orders.

Easton continues by noting that the Catholic Encyclopaedia labelled Peregrinus “the Franciscan Petrus Peregrinus de Maricourt”. Yet, the entry in that Encyclopaedia cited the work of Erhard Schlund who concluded the exact opposite. Schlund was testing the hypothesis whether Peregrinus was the same person as Peter of Arden who was a Franciscan, but ended up admitting that “there is little, if any, evidence in support” of that hypothesis. Despite this, the image of Peregrinus as a Franciscan continued being used in Francis Winthrop Woodruff’s biography of Roger Bacon. Woodruff described scenes of Bacon and Peregrinus experimenting together at Bacon’s convent in Paris, and the two of them becoming close friends. Woodruff’s intention with this image was to consider the connection between the two of them as an impetus for Bacon to join the Franciscan Order. Alas, I haven’t got a copy of this book, but Easton’s summary of it sounds like Woodruff relied very much on Hutton’s work (which is also available for free online).

As concluding remarks: the status of Peregrinus as “a monk or a priest” was important in relation to Roger Bacon’s links to the Franciscan order. However, by the 1950s the status of Peregrinus belonging to the Franciscans or to any other Order was being questioned by historians in the absence of clear evidence.

Bibliography:

Dictionary of Scientific Bibliography - https://archive.org/details/dictionaryofscie10gill/page/532/mode/2up

Stewart Easton (1952) - Roger Bacon and His Search for a Universal Science - https://archive.org/details/rogerbaconandhis027099mbp/page/n133/mode/2up

Edward Hutton (1926) - The Franciscans in England - https://archive.org/details/franciscansineng0000hutt/page/138/mode/2up

Link to the entry in the Catholic Encyclopaedia on Roger Bacon - https://archive.org/details/07470918.13.emory.edu/page/n137/mode/2up

Schlund Erhard OFM (1911) Petrus Peregrinus von Maricourt. Sein Leben und seine Schriften. Ein Beitrag zur Roger Baco-Forschung. Archivum Franciscanum Historicum 4:436–455

The reference from the entry to the German text: Hellmann, G., Rara Magnetica, Neudrucke von Schriften und Karten über Meteorologie und Erdmagnetismus, no. 10 (Berlin, 1898)