I doubt any legislator anywhere is making a vitamin A vs. global warming mitigation decision.

There is finite funding for nutrition, there is finite funding for climate, the funding is not allocated at random. So no, there may not be a particular moment where one legislator decides to split a pot of money between those two causes, but the process of creating a budget requires many decisions about spending limited resources on individual programs.

"The Copenhagen Consensus Center is a think tank that researches the smartest solutions for the world's biggest problems, advising policy-makers and philanthropists how to spend their money most effectively."

Spending money is not the only way to solve problems, but spending money can help, and it is certainly a way policy-makers and philanthropists can do good (or harm!) depending on how they act.

Can you suggest any better alternative to cost-benefit analysis for these organizations to make better decisions?

I'm curious as well to know what you have in mind as the non-financial approaches ... but on second reading you say "include, but aren't limited to, spending money" so you recognize that these countless ways do involve financial decisions. Spending money is a part, but not all, of most interventions.

You seem skeptical of the Copenhagen Consensus but I don't understand why. If the ways to address global warming include spending money, how do you decide how to spend the money? Which of the "countless ways" will you prioritize? Can an additional airplane trip be balanced by using paper straws? You question whether economists can provide estimates, but the proposals were created by "acknowledged authorities in each area," not the economists. You point out that economists have made mistakes, but don't propose an alternative to economists.