Looking for a word meaning "attempt" or "bid" or "campaign" specifically in a legal context
Solution 1:
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
argue
[ 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"
Solution 2:
What about simply "prepare"/"prep"?
- To organize or plan in advance. 2. To correct an issue or make it ready for use.
Examples:
I'm keeping a diary of these events just in case. Doing that will be part of the case I'm preparing for.
Hulk Hogan's personal attorney, David Houston, demands Gawker remove the...tape ...Gawker initially refuses, prompting Hogan to hire additional lawyers and prepare a lawsuit. Source
Here, the lawyers haven not filed a suit, are not prosecuting, appealing, defending, etc. Nothing in the legal system has been done yet - they're simply preparing for a suit. But it does necessarily mean that work is being done, and it's understood you/the firm are devoting resources and making decisions toward a wanted outcome. (I don't think many people will think one prepares for legal cases in a direction of an unwanted outcome).
While it doesn't convey explicitly that there is an "exact outcome" you're hoping for, that can be assumed in the context you use the phrase.