The closest research I'm aware of is the handful of social science workshop papers which have looked at public and lay perceptions of SRM and SRM governance. See in particular these papers:
They're mainly gauging public responses though, rather than mapping out what steps would need to be taken to increase acceptance (which is a bit of a loaded research question anyway).
Regarding your second question though, you'll find a bunch of stuff online regarding the SPICE project fiasco, and of course the more recent SCoPEx cancellation. Curiously in both cases, there wasn't a whole lot of public interest until right before the test when a couple of vocal NGOs sparked backlash.
Perhaps there's some interesting questions one might ask about how public deliberation should proceed within the context of an increasingly charged political atmosphere of environmental NGOs which vehemently oppose SRM on ideological grounds. Would more dialogue help, and how so? Should SRM researchers be more involved in dispelling misinformation? Maybe experiments just need to be done in places where there isn't as much political opposition? Do we just have to wait until climate damages get bad enough? Food for thought.
Cheers,
-A