Burton's Legal Thesaurus has definitions for pursue which might fit what you are looking for.
strive to gain; seek to attain, etc...
The Washington State Association of Counties has recently decided to move forward on pursuing legal action
County leaders frustrated with the unfunded mandates and lack of support coming out of Olympia have authorized the Washington State Association of Counties to explore legal action against the State.
In a recent famous case involving Elizabeth Holmes of Theranos Inc. a Judge ruled that Theranos investors cannot pursue class action. More information about the ruling here.
You would however need to change your example sentences to make pursue/pursuing grammatically correct in the context.
After you have started proceedings, perhaps crusade will be more appropriate
to make an effort to achieve something that you believe in strongly:
She crusaded against sex and violence on television.
Risking it all on a legal crusade which appeared in the LA Times describes a story about a lawyer who did all in his power to get a corrupt judge disbarred.
There is also the word
[ I/T ] law To argue is also to represent the case of someone in a court of law.
but it doesn't necessarily convey the effort involved to "make the case"