Let’s just say that there are many “right” answers to your questions.
In general, a typical solution would be for you to form an LLC in Delaware or Wyoming (I prefer Wyoming because its cheaper to form and maintain a company in that state), and use that LLC to do business in the US.
However, as you mentioned in your question, to justify this plan you need to have at least a few US clients, so that it would make sense for you to travel to the US to open a bank account and to spend time and money to maintain this company.