If you don't do it right with laws you can be in huge trouble for transmission of money. Just look at Charlie Shrem.
Then rather than registering the company in the United States, why don't register it in some tax havens such as the Cayman Islands or Dominica, where the requirements are lax? If the United States authorities does not allow trading of stocks, then it is OK. We can buy stocks from exchanges in some other country, where there are no KYC requirements.
The thing is I believe OP was wanting popular and stable US exchanges. A lot of possible laws to follow.
I am no expert on all countries. But if you have a country that does not require to know customer, then yes stocks there would be no issue. But I'm not sure as far as if there is any good stock exchanges with such lax regulations. I personally think you will find it hard to find a stable exchange without having some laws.
What exchange are you suggesting OP go to?