Honestly, I think it is more on the climate
scientists at this point to be vocal about it.
I think you hit the nail on the head here.It's understandable that science-illiterate, climate change-denier types fear fission. The fossil fuel industry has done an excellent job percolating their pro fossil-fuel agenda and fomenting fear of the unknown. This is the unavoidable enemy.
The only way to counter this would be for green types (Greenpeace, etc, as you say) and climate scientists to unite and promote fission. I do not think this is remotely likely, but it is the only thing that would remotely stand a chance of countering the fossil fuel industry in the battle for public mindset and votes. I would be absolutely stunned if this happened before billions are displaced due to fossil fuel-caused climate change, and I actually don't think it will happen even then. The status quo will continue as long as the fossil fuel companies remain rich... so, basically until civilizational collapse.
Simply put, fission got an extremely raw deal. It was stabbed in the back and buried by the people who should have supported it, based on their stated goals and beliefs.
Nothing can survive that.