The same problem would not occur. The reason the climate issue has been allowed to go on so far under capitalism is broken feedback loops: the people who are worst affected by climate change and the people who make the decision to emit more greenhouse gases are different people.

Under a non-capitalist system (at least from an anarchists perspective), the people affected would also be the ones to decide how much to emit -- and it's unlikely they would be as crazy about climate change.

(Not to mention that assuming "the lifestyle of today" where a small fraction of a percentage of the population does most of the consumption would be sustainable under anything other than capitalism seems like a mistake.)