What is the best way to write a court case name?

There is no "correct" way unless there is a binding set of rules for writing case names, and no such rules exist.

There is no "best" way without reference to some set of preferences. If you're trying to save keystrokes, the first is best; if you're trying to fill as much space as possible, the last is best.

If the question is, "How do American lawyers write court case names," the most widely accepted reference would be The Bluebook, which would render the case as Dobbs v. Jackson Women's Health Org.

If the question is, "How do American journalists write court case names," the most widely accepted reference would the the AP Stylebook, which would render the case as "Dobbs v. Jackson Women's Health Organization."

Australian courts (to the extent that the title page of the judgement contains the correct citation) and universities apply the Australian Guide to Legal Citation. A free-to-view version is here.

US case law is dealt with in section 25.1:

So, for your example: Dobbs v Jackson Women's Health Organization, 597 US _ (2022).

AGTLC supports Short Titles so the first mention could be Dobbs v Jackson Women's Health Organization (‘Dobbs’) and subsequent citations would be Dobbs.

AGTLC is a footnote-based reference system but you can use the name of the case in-line but would need to reference the other details in a footnote. Something like:

We now consider the decision in Dobbs v Jackson Women's Health Organization (‘Dobbs’). 23

23 597 US _ (2022) (‘Dobbs’)

In a rare move, the US Supreme Court has overturned its own precedent. 23

23 Dobbs v Jackson Women's Health Organization, 597 US _ (2022) (‘Dobbs’)