This is not true. Harry does not have a case in the US today.
The case is between Heritage Foundation, a right wing think tank that is trying to the Harry deported from the US, and the US Department of Homeland Security.
It is not Harry's case as he is not a litigant, but relates to his visa application.
Heritage has asked their freedom of information request to be "fast-tracked". citing press interest in Harry's visa.
The DHS's written opposition to the fast-track request described the press interest from British publications as “niche” and it was not persuaded that there were exceptional circumstances that made an urgent response necessary.
In other words, they argued that government processes cannot be speeded up to satisfy the hunger for tabloid headlines in another country.
As for your comment:
I wonder if that’s why she is staying away, having had her knuckles severely rapped by a British judge?
Meghan is not a litigant in these cases. I imagine that is why she is "staying away". I also imagine that winning the case more than made up for having her "knuckles rapped".