In my area of the US, the business license is for where you're located. Unlike contractors, who often need a business license in every city where they build, I only need a business license in the city where my office is located.
Most states will require a firm registration regardless of status/structure - sole proprietor, (P)LLC, PC, Inc. - they want to keep track of who is offering these services. This is separate from the individual PE license/registration.
As XR pointed out, most boards won't really go looking for people who are violating this, but if you make a mistake that gets the board's attention (could be a complaint from a dissatisfied client, an error that results in an incident that gets investigated, an overzealous building official who looks you up on the board's website to verify your credentials, etc.) then they will happily slap a disciplinary action and fine on you.
Then, the secretary of state of wherever you want to work is going to need to issue you a certificate of authority to do business in the state as a 'foreign entity' (not just because you're from a different country - I'm a Virginia PLLC so I'm a foreign entity in North Carolina, 6 miles away). To get that, you'll need a registered agent with a mailing address in the state. That doesn't come free.
Business and professional licensing compliance in the US can be an absolute nightmare. Maintaining a personal PE license, a firm registration, a CoA, and a registered agent in each state where you practice, all with different continuing education requirements, different renewal schedules, different reporting requirements, and fees for all of it.