I don't think there are many circumstances where someone just 'wants' to kill a polar bear, or any bear for that matter, but there are several reasons that it would/could be necessary, maybe even merciful. Polar bears are some of the most aggressive bears there are, but thankfully they don't typically inhabit areas where there are lots of humans.
In cases of unintentional contact, it might become a personal defense situation. But again, the rarity of those contacts lead to very seldom circumstances of a required killing of the animal.
Then there are also circumstances of necessity ...... for example a sick or injured animal. This could arise from the aftermath of two old boars fighting and one being critically hurt in that fight. Or an old age situation leading to probable death over the coming winter, or I've even seen times when parasites and/or disease in a herd would dictate culling.
A (not very likely) situation of animal population control where too many bears may congregate in one particular area due to many factors. This can occur when human encroachment into a natural bears habitat causes a desirable attraction for the animals by the food, scraps, and waste of those humans.
And then there are some special allowances for certain natives of the north. These allowances are generally very limited and very specified, often based on tradition, excess population , general location, etc, of the animals.
And generally speaking, as limited as 'trophy' hunting for polar bears is, it still does happen but to a very limited degree. It's my understanding that any animal permitted for 'trophy' hunting is culled very carefully based on many factors. Some of those factors even mentioned above.
This topic brought up a good question regarding the taking of a bear for any of many reasons, so I looked around a little for more info. I found this short publication and thought it worthy of providing it here:
Find answers here to frequently asked questions about polar bears, sea ice, climate warming and more..
polarbearsinternational.org
And let me add this ........... I personally am not a 'trophy' hunter of any sort but I am a hard hunting and ethical sportsman, and over many years have taken a few animals I was proud to display on my wall for one reason or another. But that's just me. I also recognize there are many legitimate reasons for 'trophy' hunting in some cases. Many times the real benefit of any particular 'trophy' hunt is for the animals themselves, whether individually or overall for the species.
And lastly, I absolutely will not tolerate any illegal or unethical hunting/trapping of any kind. I'll turn in a poacher in a heartbeat with absolutely no heartburn about it. I'll even get really upset when someone unintentionally calls a 'poacher' an illegal hunter, or a 'night hunter', or any other name using the word 'hunter'........ they are NOT hunters in any sense of the word. They are low life, scumbags, not worthy of anything even close to descriptive of a hunter.
But again .............. that's just me!!!!