Vjylku

In his answer to a recent question on whether there had been a Viking Exchange between Europe and the Americas, LangLangC brings up that the King of Norway offered a polar bear to Henry the 3rd in 1252 (1251?), and that another such polar bear made its way to the Sultan of Egypt via Emperor Frederik earlier that century.

The official Historic Royal Palaces website adds this nugget:

Although it was kept muzzled and chained, the bear was allowed to swim and hunt for fish in the Thames.

A collar and a ‘stout cord’ were attached to the bear to keep it from escaping.

Insofar as I understand, male polar bears range from 350-700kg and measure 2.4-3m, and females are about half that size, and I'm not aware of any reason why that might have changed much in the past thousand years. Its medieval zookeepers would have been about an inch shorter on average than we are today.

Even in the best of scenarios (a small female) polar bear is comparable in size and weight to a typical adult male black bear, so I'd imagine they were captured and tamed while still cubs. Still, I'm somewhat struggling to make any sense of how one would go about with muzzling an adult polar bear without tranquilizer, except perhaps one (dead or injured) zookeeper at a time.

Is there any documentation anywhere on how they were handling the beasts? More specifically, on how they'd get the muzzle on without anyone getting injured or worse?

One primary source I can trace for this is appears to be the Calendar of the liberate rolls

On Sept 13, 1252 it states:

To the sheriffs of London. Contrabreve to let a white bear, which the
king is sending to the Tower to be kept there, and its keeper, have
4d. daily for their maintenance so long as they are there.

On Oct 30, 1252 it states:

To the sheriffs of London. Contrabreve to let the keeper of the king's
white bear, which was lately sent to him from Norway and is now in the
Tower of London, have a muzzle and an iron chain to hold the bear when
out of the water, and a long and strong cord to hold it when fishing
in the Thames.

Lucy Inglis, in Georgian London: Into the Streets implies that the animal was not full-grown when it arrived, but there is no source for this. If true, then it may have been possible to tame it to some extent as in the (modern) case of Mark Dumas and the 800 lb Agee (there's also a video):

Agee is a 60-stone (800lb) polar bear that they've managed to train to
star in high-budget TV adverts and movies. She's also often seen
messing about with Mark, giving him bear hugs, accepting loving kisses
and even swimming with him.

An accessible source for the story of the "White Bear", which is presumed to be a polar bear, is to be found in Thomas Maddox's The history and antiquities of the Exchequer of the kings of England. This also includes the instructions to the Sheriffs of London to build a house at the Tower of London for the King's elephant:

"The Sheriffs of London were commanded to supply four Pence per Diem, for the Maintenance of the King's white Bear and his Keeper in the Tower of London The fame Sheriffs were commanded to provide a Muzzle and an Iron Chain and a Cord for the King's white Bear in the Tower of London (a); to build a little House in the Tower of London for the King's Elephant (b); and to find Necessaries for the King's Elephant and his Keeper in the Tower of London (c)"

(a) Rex Vicecomitibus Londoniae salutem. Praecipimus vobis, quod cuidam Urso nostro Albo quem mittimus usq; Turrim nostram Londoniae ibidem custodiendum, & custodi ipsius, singulis diebus quamdiu fuerint ibidem, habere faciatis quatuor denarios ad sustentationem suam. T. Rege apud S. Edmundum xiij die Seplembris. Liberat. 36. H. 3. m. 4. [Hanc instantiam, & sequentes sex septemve, mecum communicavit Vir in me amicissimus Georgius Holmes Generosus, Antiquarius.]

(b) Rex Vicecomitibus Londoniae salutem. Praecipimus, vobis quod custodi albi Ursi nostri, qui nuper missus suit nobis de Norwagia & est in Turri nostra Londoniae, habere faciatis unum Musellum & unam Cathenam ferream, ad tenendum Ursum ilium extra aquam, & unam longam & fortem Cordam ad tenendum eundem Ursum piscantem in aqua Thamisiae; Et custum &c computabitur &c. T. R. apud Windesore xxx die Octobris. Liberat. 37. H. 3- m. 15.

(c) Rex Vicecomitibus Londoniae salutem. Praecipimus vobis, quod de firma Civitatis noftrae Londoniae, sine dilatione construi facialis apud Turrim noftiam Londoniae, unum domum longitudinis xl pedum & latitudinis xx pedum, ad Elefantem nostrum; provisuri quod taliter fiat & ita fortis fit, ut cum opus fuerit ad alios usus apta & necessaria. Et custum &c computabitur vobis ad Scaccarium. T. R. apud Westmon. xxvj die Februarij. Liberat. 39. H. 3. 7n. 11.

To the best of my knowledge, we have no record of any injuries sustained by the keeper responsible for the white bear, so I think we can assume that, however he attached the muzzle in practice, the bear was fairly amenable, and the keeper good at his job!

Assuming you can get it into a cage somehow, get it to stick its head out (wave a fish in front of it?) through something resembling a pillory/stocks attached to one of the gaps. Now clamp this shut. Optionally, capture its snout with a loop on a pole. It can't move much now it's in a neck-lock, so it should be easy enough to get the muzzle on now.