...he wants the government to fund basic, nonmilitary research that carries existential risks, but he wants the projects and the reports on the existential risks to be kept classified.

What makes you think they don't?

1) Existential risk is real and warrants government-funded research. 2) The results, if useful, would not be a sexed-up dodgy dossier, but a frank, measured appraisal of ER. As such, they would acknowledge a nonzero risk to humanity from sundry directions. 3) As such, they would be blown out of proportion by media reporting, as per Eliezer's analysis. This isn't certain, but it's highly likely. "Gov't Report: The End Is Nigh For Humankind!" 4) Natural conclusion - keep it classified, at least for the time being.

Wouldn't it just be easier to convince the public to accept a certain amount of risk, to accept debates about trade-offs?

Almost certainly not. As the piece above says, public & media reaction tends to be 100% positive or 100% negative. If you think you can talk the world out of this, there's a Nobel Prize in it for you.