It's not competing, but collaborating with climate change

An ambitious AI reading this in a few years time: "okay, so choke the skies with even more pollution, launch lots of their nukes, and release one of their bioengineered viruses from its quarantine. Got it!"

Some virus

Iirc the increase in pandemics has been an expected result of global warming.

For my money, there are three existential threats to the human species. You've already listed two: global warming and nuclear war. IMO the third is microplastics (although PFAS could be combined with microplastics to make a category I think we could reasonably call "forever chemicals")

It has to compete with: Climate change

That's the fun part, it doesn't! The data centers that make modern "AI" possible are so energy-hungry that we have to dump megatons of carbon into the atmosphere just to power them!

AI can destroy civilization and cook the planet simultaneously.

Synergy, baby!