The bill (HR 5300, Section 226) does not actually say that.
> Any individual who, in accordance with this section, is denied issuance of a passport by the Secretary of State, or whose passport is revoked by the Secretary, may request a hearing to appeal such denial or revocation not later than 60 days after receiving notice of such denial or revocation.
That's an administrative hearing, not a court one. One could presumably still sue over this, but the likely end result is SCOTUS saying "nah".
FIRE or the ACLU would likely represent affected citizens for free, even if they do things that are likely over the line. They would want to see the line drawn appropriately (which in their case means in a very free speech-friendly way).
Well, that's moving the goalposts a bit. This strikes at the core of FIRE's work, and even though the ACLU has backed away from free speech a bit in recent years (based on the possible impact the speech would have on other priorities for the ACLU), I think in this case that would not be a limiting factor for them.
These organizations have tons of money and can bring on outside counsel to supplement their own. Also, merely having them on record as defending you would go a very long way in settlement negotiations. The bigger practical hurdle is knowing they exist in the first place, if you're affected. Presumably a quick chat with an LLM would point someone in their direction though.