What is the likelihood of obtaining net neutrality through the courts? I.E. Cloudflare sues -> judicial process -> decision that establishes a "right to access"?
Likely 0% chance. The court cannot just go off and make up its own laws because it wants to, all it can do is decide how existing laws should be applied.
It's true that courts can't make laws, but they have shown a lot of leeway in the past of creatively interpreting laws (i.e., using the Interstate Commerce clause to say a farmer can't raise a particular crop to feed to his own animals). Could existing laws that prevent monopolies from unfair business practices be applied here?
Unlikely through the courts, although state utility regulators (i.e. the California Public Utilities Commission) might take interest. Democrat members of the FCC also might be interested.