IMRhys published a detailed article about setting up a company from outside the U.S. It's very detailed and not the easiest but take a look and see if it is doable for you. It includes such things as incorporating, getting a U.S. mailing address and U.S. banking. All without needing to be in the U.S.
He says it is possible to get U.S. banking from a foreign country with
Silicon Valley Bank. It's written for starting a dropship business but it could be used for any business.
https://www.imrhys.com/start-a-united-states-dropship-business-from-overseas/
Silicon Valley Bank
https://www.svb.com/
My USA Corporation has an article as well and offers other options besides Silicon Valley Bank. They also offer services to help you set up a corporation in the U.S. I have never used them.
https://www.myusacorporation.com/articles/us-bank-account-for-non-residents
I'm not associated with any of these sites and have never used their methods but I had the links bookmarked from when I was doing research.
These methods can get expensive because they require incorporating in the U.S. and lots of faxing/mailing of documentation.
You could search
Google as well to see if there are other articles or service providers but I did that a while ago and these are the best resources I had book marked.
I did find this discussion thread on
Reddit but I haven't gone through it to see how useful it is:
https://www.reddit.com/r/startups/comments/2gcxbj/question_opening_a_bank_account_in_usa_from/