Even climate scientists don't argue it'll be extinction level, it'll just be really bad for some regions, great if you live in Siberia and suddenly Canada and Russia have shipping lanes in the arctic and are growing tomatoes. Calm down.
The Great Filter doesn't refer only to extinctions. Anything that makes it highly improbably for a civilization to advance from our current level of technology to significant space colonization would be a Great Filter.
Climate change itself perhaps not. I'd argue there's quite a bit of precedence to believe that the way humans will react to these changes could lead us down that path.
The same as its always been. A handful of people on top of some arbitrary social hierarchy in charge of the lions share of available resources. I believe this type of social organization is in our innate behavior like how insects form colonies and fish school.
The great filter is something with odds around 10^-30 of surviving.
It's hard to find any convincing argument that climate change will destroy our civilization with anything near 1/2 odds. It's a completely different ballpark.
I think it’s a definitional thing: “a great filter” assumes that life is basically everywhere and yet one specific something (or perhaps as many as “a few”) almost always prevents it from taking over a significant fraction of its future light cone.
If there are 10^30 stars in ours, the lower bound on the odds of surpassing it have to be close to that.
Personally I expect there to be many smaller filters rather than a few great filters — 30 things each with a factor of 10 has the same effect, after all.
Personally, abiogenesis by itself looks like something on the 10^-60 to 10^-140 range, so it fully explains everything.
Anyway, there are indeed a few things in our past that look like a 10^-10 odds filter, like each time our genetic encoding changed to increase our codons and the oxygen catastrophe. There are more that look like merely "almost impossible to survive" (I'd guess some ~10^-4 filter) like large asteroid impacts, but not nearly enough to make a difference.
Some will survive but many will die as well. Our modern society has increased the carrying capacity of our planet. As we see climate change disturb food production we will see this carrying capacity fall. If this leads to the collapse of stability in regions the carrying capacity will fall further.
Climate change is a natural fact of our planet, it's been happening since its inception and will continue. Is this current push towards renewables our first forey into terraforming?