The better question is whether it is wise, and in this particular cae I suspect it may well play to their favour - it's a rare latin nerd that fails to chuckle at a bad latin quip or three.
After all, it's not like they led with Pēdīcābo ego vōs et irrumābō from Catullus 16.
But not in the US.
[I believe the GP is from outside the US]
What gave you the truly outrageous idea that justice department has the capability to "quash" lawsuits such that they can't be appealed?
If you actually read the questions presented in the petition for writ of certiorari (https://www.supremecourt.gov/DocketPDF/22/22-293/238740/2022...), it's more anodyne:
> The questions presented are:
> 1. Whether an officer is entitled to qualified immunity for arresting an individual based solely on speech parodying the government, so long as no case has previously held the particular speech is protected.
> 2. Whether the Court should reconsider the doctrine of qualified immunity.
So, yay, it's basically a qualified immunity case, where the infringement in question is an unhappy local government office patently violating someone's First Amendment rights. (Sadly, not the first or last time this is going to happen.)
[1] https://www.loweringthebar.net/2012/09/all-briefs-should-now...