I agree that the English form of Sverre's name is undoubtedly now Sverre, but it's just a little funny to see Britannica doing what they always do -- smugging the Norwegians by using a more authentic form of his name than they commonly do. Academic humor, I suppose ;)
Historians writing in English have been erratic about conventions for anglicizing unfamiliar foreign names. I've been collecting some examples for Sverre. For example, Sverus (1854), Sverri (1899), Sverrir (1920), and Sverre (1922). The modern fashion, of course, is to get as close to the familiar form in the relevant modern language as possible.
I thought it would be interesting to know what the English form of Sverre's name was in his own lifetime. There had to be an English form. Norway had strong commercial ties with both England and Scotland during Sverre's lifetime. Also, Archbishop Eystein Erlendsson, who opposed Sverre, was in exile in England for 3 years (1180-1183). And, King John of England sent mercenaries to help Sverre 1199/1200.
The problem is that this is a period before people were writing in the vernacular. Everything original is going to be in Latin. But even with just Latin, it's often possible to make a good guess about how someone pronounced the name they were Latinizing.
My first thought was to check Diplomatarium Norvegicum. There are many documents from Sverre's time there, but I didn't find much from England or Scotland. Most of what's there is about other people.
I did find a 1200 letter from Pope Innocent III to the English archbishop telling him not to accept any more gifts from Sverre (17:1223). Not really what I was looking for because it was written in Rome, not England or Scotland. It calls him Suerius.
http://www.dokpro.uio.no/perl/middelalder/diplom_vise_tekst.prl?b=1...=
The closest to what I was looking for was an entry in 1282 for a letter from Sverre to King William the Lion of Scotland (19:69). I couldn't find the text, but I wasn't as interested in that because it was written in Norway, not Scotland. But, the index entry seems to have been written in Scotland. It says, "[Carta] Sweri Regis Norwagie" (Letter of Swerus, king of Norway). Sweri is a possessive form, probably for Swerus
http://www.dokpro.uio.no/perl/middelalder/diplom_vise_tekst.prl?b=1...
My other thought was to look at William of Newburgh's Historia Rerum Anglicarum (12th century). He also calls him Swerus. And, for what it's worth, he says that Sverre's seal had the incription "Suerus Rex magnus, ferus ut leo, mitis ut agnis" (Suerus great king, fierce as a lion, gentle as a lamb).
https://books.google.com/books?id=k7VCAAAAYAAJ&dq=William%20of%...
So, we know that his name in his own language was Sverrir, which he himself Latinized Suerus. His name in Italian was something that could Latinized Suerius. And, his name in English was something that could be Latinized by someone in Scotland and someone else in England as Swerus. This isn't much evidence because there is a bit of variation in the way names are Latinized in different documents, even with the same scrivener, but the fact that two different English-speaking people Latinized his name Swerus is probably significant.
Finally, I asked a friend who knows much more about Scandinavian languages than I do about the English form of Sverre's name during his lifetime. He promised to look it up for me if he gets a chance, but he thinks it would have been analogous to the relationship between the names Sven and Sweyn.
His best guess was that the Middle English form of Sverre's name was probably Swerra or Swera. After I told him what I found in contemporary documents, he was even more confident because the obvious way to Latinize Swerra would be Swerus.
Just a fun digression from Erica's more serious point -- poor Sverre needs to have an English display name.