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!
It may also be worth considering the case of Knut, the polar bear that was hand-reared at Berlin zoo. In that case, modern attitudes and concerns for the well-being of both Knut and his keepers meant that the bear's interaction with human handlers was diminished as he got older. I doubt those factors were as important in the thirteenth century!
The example of Knut shows that it was possible for polar bears to become acclimatised to being around humans. In which case, it would probably also have been that much easier to get it to wear a muzzle.