Some attorneys will work on a fixed amount paid up front, but yes, legal aid is not cheap.
In cases where you stand to win money from damages, they may also work partially for a cut of those damages, but that doesn't apply here.
One of the people on law-twitter also noted that the lawfirm nu-TSR is using appears to be primarily an insurance lawyer. Generally, you want to have legal representation that is specific to your field, since IP law is really complex. That doesn't mean the "wrong" type of lawyer is not helpful but it's much less helpful than someone who is an expert in the specific field.
The two times I've had to retain a lawyer in the US, we went to lawyers that were specifically specialized in a certain kind of case. In one example, the ONLY type of cases they did was what we were asking about.