There are ways around the issues you've identified.
1. LLCs/INCs get their own tax ID number (EIN), as can several other types of entity, including foreign entities - for example I have a US bank account for a US branch of one of my NZ companies and all I needed was a passport - personally, I'm not eligible for an SSN but I am eligible for an ITIN - and depending on how legal you want to keep it would depend on what you might want to do next as far as opening a US account went.
2. On the other hand, you could maintain some privacy by using a nominee for your LLC, open the bank account in the US, if you decided to open a bank account at all, but it may not even be entirely necessary, as there are plenty of prepaid and/or virtual options you could use both in and outside of the US...
From what I understand there are a lot of poorer people who don't even have bank accounts in the US, so I guess one would need to get familiar with how they are able to move their money around, but the first thing that comes to my mind is those prepaid cards bearing the Visa/MC logo. Since they seem to be able to be purchased anonymously and I'm guessing you can use it for almost any kind of domestic transaction (like if your ISP allows you to pay by card online just by logging in to their billing portal - I know some do but I can't speak for all) - all you need to do is have enough money on the card and presto, bill is paid on behalf of XYZ LLC.
At the end of the day you'd probably just have to watch out for the law firm accidentally having a data breach/leak and your anonymous involvement in XYZ LLC becomes exposed. Hopefully though by then the statute of limitations will have passed.