The American Libertarian Party would probably say that the court system is paid for by government revenues from taxes and tariffs like the Constitution says.

Anarcho-capitalists would say that it's paid for by whatever revenue stream Courts, Inc[1] develops; they wouldn't presume to enforce a single business model, you see (I assume).

In the case of "left libertarian" thought, it again depends on the particular society envisioned - I don't really have enough understanding of the various models to give a clear answer, but most left libertarian utopias are even further from our modern situation than the right libertarian ones. Of course, most wouldn't have the financial infrastructure at the heart of this particular case anyway.

This is not to say that I have any high degree of confidence that any of the models discussed or glossed over above would actually work (well or at all), but disagreement does not sanction misrepresentation.

[1] Although I think some of them are opposed to limited liability, so it might or might not be "Incorporated"...