Short answer
Although we don't have much detail on how this large animal was handled beyond "muzzled and chained" and "collar and a ‘stout cord’", it is likely that the white bear was relatively tame and that its keeper was experienced at handling the animal.
The experience of late 19th century / early 20th century circus keepers was that polar bears were not difficult to tame but that keepers had to recognize the personality of individual bears. Nor, it seems, did they necessarily need tranquilizers or any other modern 'aids' which would not have been available in the 13th century.
Details
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, 1253 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.
According to Nigel Jones in Tower: an Epic History of the Tower of London, the white bear came with a Norwegian keeper. There are many stories in Icelandic texts concerning polar or white bears as gifts and / or in the company of humans, of which Audan's Tale is probably the most famous. Although the historical accuracy of some of these stories is questionable, it is clear that the idea of gift-giving and managing polar bears was one familiar to Norwegians and other Norsemen.
King Haakon IV of Norway (reigned 1217-63) was near contemporaneous with Henry III (reigned 1216-72) and had long maintained close relations with the English king. The bear was not the first gift exchange: Haakon had sent Henry an elk in 1222, and Haakon's seal had been given to him by Henry in 1236. It would therefore not be unreasonable to suggest that the Norwegian king chose his gift with some care.
Further, the bear was not the first dangerous animal in Henry's menagerie - he also had (or had had) lions and leopards - so there may have been someone with experience in handling large wild beasts should the Norwegian keeper have required assistance.
Lucy Inglis, in Georgian London: Into the Streets implies that the bear was not full-grown when it arrived. Although she cites no source for this, it is plausible as there are tales of cubs being brought to Norway from Iceland.
Michael Engelhard, in Ice Bear: the Cultural History of an Arctic Icon, relates the experiences of late 19th / early 20th century keepers. It is entirely plausible that Norsemen would have known much of what follows:
Training, for bears born in the wild, began with their capture...,
with the preliminary step of taming by animal keepers.... Most traders
and circus men thought the polar bear, if taken young, was easily
gentled.
Trainers preferred bears that were about one year old.
Polar bears, though, can be unpredictable and care has to be taken not startle them. They can also get irritable in warm temperatures (so a dip in the Thames wouldn't have done any harm). Englehard adds that:
Trainers found that it was important to get to know the temperament
and idiosyncrasies of each animal. Like other people familiar with the
animal, such as Native hunters,...they speak
of differences in the behavior of individual bears;... Trainers... developed a repertoire of
techniques for different bear characters, from the “unmotivated
toddler” to the “obstinate” or even “volatile” “teen.” ...A keeper spent ... time “to educate
them out of their savage state—by contact, kindness, sugar and fruit”...
There is also the (more 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.
