That isn't the way biology or evolution works.
The moths of Britain during the industrial revolution became darker. The simplistic reason is "because of the pollution of coal soot." The more accurate (but still simplistic, yet useful) answer is "because lighter moths were easier to hunt in the changed environment so there were fewer light moths to pass on their genetic variant, leaving only those moths with darker coloration." In other words, the moths didn't actually change so much that a tendency to have a darker color, which already existed, persevered over a lighter color.
I need to emphasize this point. The moths didn't "evolve" a new color (not quite true, but hold the thought). The ability to be colored darker already existed in the moths' genome. What happened is that an easily hunted variation of the moth was exterminated, leaving a more difficult to hunt variation of the moth to reproduce. Curiously, a century later and after untold volumes of laws and the invention of technology brought about cleaner air, the number of light-colored moths is increasing... because the darker color is now easier to hunt than the lighter color. This is one of many ways that evolution operates, but it's the short-term version (sometimes called "adaptation") vs. the long-term version where the genome itself changes.
But that's not what you're explaining. You're explaining an intentional change due to detected variations in the environment, not a shift in the dominance of one variant over another due to a change in the environment that favors predation of one variant over another. So when you ask, "is this feasible?" the answer is "not according to any science we understand today."
But let's ignore "is this feasible?" (which isn't what we do well) and focus on "is there an example from Real Life I can use to model something believable?" (which we do very well)
Let's look at those British moths more closely. Specifically, let's look at the life cycle of the Peppered Moth, which isn't dissimilar to pretty much any moth or butterfly, but it's useful for our purposes.
Peppered moth eggs hatch during mid summer. Larvae (caterpillars) feed on the leaves of birch, willow, and oak trees. The larvae look much like a small branch. Having a body that looks like a stick helps the larvae hide from predators. The larvae can even adjust their color from brown to green to best match the branches they are feeding on.
Cold weather is difficult for insects. To avoid death, peppered moth larvae change into pupae (cocoons) for the winter. In April and May the pupae open to reveal a new adult moth. These adults will lay eggs and die by the end of summer. No peppered moth lives for more than one year.
Your creature has variations that are happier in (e.g.) hot weather than in cold weather and vice-versa. Just like humans, if you think about it. But what if your creature, when it encounters an environment it's not "evolved" to survive (i.e., that the variant isn't disposed to prefer — similar to the larvae when cold weather sets in), doesn't just hibernate, but changes into a pupae? Unlike moths, they can do this (perhaps) as often as is necessary, though in the course of "natural life" this is highly unlikely to occur more than twice: once when changing from juvenile to adult, and once again when the new environment demands it. After that the creature (probably) hasn't the strength to do it again, but that doesn't matter because by that time (save for "special circumstances" where things happen unnaturally quickly) they have reproduced and passed on the ability to once again adapt to a new extreme environment.
Using this rationalization, it would be unbelievable for such a creature to "evolve" fur when moving from a hot environment to a cold environment. That's changing the genome. But it would rationalize the creature developing a (more) dense fatty layer to preserve heat. In other words, we're looking for a way for the creature to fundamentally remain the same creature, but to enhance or diminish a pre-existing capability to better handle the changes in environment.
Remember, changing the genome (suddenly growing fur where the ability to grow fur didn't previously exist) takes hundreds of thousands to tens of millions of years. But shifting from (using a metaphor) a mostly-bare-chested human male to a mostly-hairy-chested human male using a method like what the Peppered Moth uses... that's believable.
Note: An astute reader may point out that the Peppered Moth changes from a caterpillar to a moth, the one having wings and the other no, so why can't such a fundamental difference also exist in the OP's creature? The problem is that, to my (admittedly limited) knowledge, there isn't an example of such a creature changing back from a moth to a caterpillar. It's a one-time one-way street. What I'm suggesting is that we use the idea to rationalize a (most likely) two-time one-way street. The creature can bundle up to increase or decrease what it already has, but it can't materially change from one thing to another on the second pass. Hopefully I've made this clear. Maybe not. It's getting late.