Yes, climate change is a serious problem. But is it a monetary problem or a fiscal problem? Obviously it is the latter.
Central banks are supposed to be independent and run by unelected technocrats in order to control inflation. That's typically in their charters.
Governments could delegate climate action to central banks as a political decision. But they have not.
Until they do, should central banks take it upon themselves to act? I would say no, for two reasons: (1) They do not have the political accountability to make fiscal/redistribution decisions. (2) By acting politically they risk losing their independence, thus risking their mandates to control inflation.