I think this question is rather terrestrian. In that it's asking for a lot of stuff for an alien creature that only really makes sense in context of human-like or earth life. And kind of views the world at the level of organism. First off, life forms are themselves carrying around their own biome. Like all of them, it's kind of what a multicellular lifeform is. It's a lot of different cells that work together in important and cooperative ways. I cannot digest most the things I eat but there's a bunch of stuff in my gut that can turn that into things. Also, my body heats itself up because that's the chemistry runs faster in the warmth, a lot of that chemistry is done by mitochondria which is a different critter than my cells are, or I am.
So note, everything you're asking for is perfectly reasonable and to some degree stuff that life does well. We live in an atmosphere that is one filled with oxygen, one of the most corrosive substances known as communities of cells, and in cooperation with bacteria etc. And things live in habitable zones. That's why those zones are habitable, things live in them. Extremophiles are often more the rule than the exception. Life is quirky like that.
The only thing you need is energy. You have a bear sized creature that is lives a nomadic lifestyle. It needs energy to perform this movement. And generally it will tend to need a reason to be nomadic. Animals don't waste energy going places if they make their own food. Trees are not known for being the quickest creatures on the planet, in part because they make their own energy. But, while some trees are bear sized, they are absolutely not nomadic. They move a bit towards the sunlight but it's certainly not very fast.
Bears can easily gobble up 20,000 calories a day, sometimes as much as 100,000 when the energy is plentiful and then hibernate when it gets cold. So what exactly on your planet provides this massive amount of energy, in some particular place, that these animals must go to in order to survive? And if there's a bunch of energy out in the snow, why aren't there other creatures that eat it?
You could certainly make them basically as large as bears and active as rocks sucking down the energy of the space radiation, because they live in a communal relationship with some crystal fungus but that's not really nomadic. Or you could make them kinda active like bears but then they need the alien equivalent of scarfing down a whole crap-ton of fish, or a consistent diet of some other energy source calories a day, which then wouldn't really require the nomatic part of it. They could very much graze some kind of radiation consuming moss like stuff for energy, or rather take it with them in some fashion.
But, there'd be some reason the radiation consuming moss would need to get around on a bear like creature. Plants, because they make their own food aren't the most cooperative of species. You might be able to get 3000 calories a day from intense space radiation, but that amount would certainly warm the place up pretty significantly, if it had to fall on the surface of a bear. Is there something else, like methane snow a bunch of oxygen around? If you have chemicals that can release a lot of energy together, you could certainly have some biome help with the conversion to usable energy.
--Edited to add:
1 Cold resistance. - This isn't a thing. If you live optimally at these temperatures they aren't cold for you as an organism. We like the temperatures our bodies maintain because our chemical reactions go well at those temperatures.
2 A means to cope with their poor atmosphere. -- This isn't a thing. If you live with a thin atmosphere then this would be poor for humans, since we need to breathe oxygen and don't do well with solar radiation, but if you didn't evolve for that then it's not something that matters.
3 Adaptations to survive solar radiation. -- This is a slight thing. In theory high energy stuff can break complex things but oxygen is one of the worst and most destructive things and we breathe it like a boss. And this sounds like a lot of nummy nummy energy.
4 An inner biome to produce food and recycle waste -- All bodies are doing this. That's life's jam. If it's something you can recycle bodies do this. The producing food is a problem. Since you still have to obey the laws of thermodynamics.
You could make it hibernate for like 25 years when the radiation absorbing moss covering its body builds up enough energy reserves to power its bear to the spawning grounds, and all the bear/moss cooperative critters go and have bear babies and moss babies. Or have some additional energy around. Maybe the radiation absorbing moss stuff is optimally adapted for the pole, and can't breed that far south, and only by riding a bear-like creature south and then north could it get successful.
Calling something inhospitable is just wrong without context, things only need energy, beyond that life finds a way. Nothing else from the planet lives there because nothing else can really harvest the energy (whatever energy is has). But, if stuff lives there it's both hospitable and habitable. Things like radiation and heat and cold and atmospheric pressure seem inhospitable because we, as humans, can't live there. An alien that lived there would have no issue at all.
Ps. I used moss as a shorthand for the only radiotrophic stuff I know about from Earth. That went and very quickly took extreme radiation of a nuclear meltdown and turned it into energy. Because, again, these aren't problems. Extreme radiation like that isn't a problem, it's a food source.