I'd love to be proved wrong on this.
If you run into problems, buy an off the shelf Delaware corporation online, with a registered agent, and create the account for that. Bonus, depending on where you do business you can potentially save quite a bit of tax by structuring things properly that way.
I've held personal and business accounts with Chase and TD Bank, as well as trader accounts with a couple of brokerages, and the documentation requirements were pretty basic in all cases. I've looked into opening accounts at other banks at various points too.
In some cases the easiest alternative is to simply buy an off the shelf US corporation, and have the company that sets it up register a US bank account for it. That's certainly often less paperwork and hassle than dealing with a personal account, especially if you set up a Delaware corporation.