If you truely care about your privacy I guess best would be to go with a nominee shareholder and director.
His information would be visible in company register, but yours won't be shown.
Most customers probably wouldn't care neither if you have an LLC or a British LTD. As most of them wouldn't see it straight forward besides they check imprint or whatever.
Your nominee would have to do the KYC for your High Risk PSPs though, which might be a little difficult depending on how "close" you're a with him or her / or how good your connection is to the nominee.
Rotating PSPs / miscoded solutions might work as well depending on how high your volume will be.