I'm sure you've checked this already, but RE: the removal policy at Courtlistener:
"
If you would like a case removed from the results of the major search engines, please send us a written request to the address on the contact page. Please include the pages you would like removed. Upon receiving this request, we will generally block search engines from indexing pages on our site by using the robots HTML meta tag and the x-robots-tag HTTP header."
They don't really state what, if any, conditions or factors are used in them deciding whether to deindex the page(s) or not. Almost appears as if you only have to ask...
While the pages would still be on their site, at least they should stop popping up in the Google search, as you've indicated being the problem.
To get them to remove web pages entirely, you'd need a court order. An attorney may be able to find a reason to get that done(?). Dunno how much your client is willing to pay to make that SE result disappear.
RE:
"We will not remove any public document from our database without a court order. If you want information deleted from our site, your only recourse is to get it deleted from the public record by contacting the courts that issued the opinion, or to otherwise obtain a court order. In such a case, you should likely contact the court directly."
Source:
https://www.courtlistener.com/terms/#removal
And along those lines, they are a non-profit that accepts donations...
https://www.courtlistener.com/donate/