Unfortunately, Greencard can't be used like a tourist visa. It is for having a life on US soil so the steps to take for maintaining her status is to live in the US if that's the route you go.
Regarding, the denial of her tourist visa I think she couldn't this time around convince them that she will return back to her home country after visiting.
I get that your mom doesn't want to live here. But will she be willing to live for 5 years, get US citizenship, and go back home?