Do not automatically assume that you do not pay tax personally on UK dividends!
The company pays corporation tax at 21%, for which you personally get a 10% "tax credit" on your dividend but you have to pay income tax on any amount that pushes your total income into the higher rate tax band.
In fact, the only real reason to take a dividend from a limited company is to avoid paying NIC, but the higher Corp Tax rate virtually covers that anyway.
IR35 put paid to dividends being better than PAYE, certainly for contractors.
If it's a sole proprietor operation, could you not elect to have it treated as a self employment, declare your earnings on your 1040 and waive the self employment tax with the certificate? Thus it's transparent, not separate from you as an entity, and the "corporation tax" is just in lieu of income tax as the above IRS blurb indicates?
I'm looking into doing something similar but am, quite frankly, completely put off by the complexity of US/UK tax law. I'd rather stack shelves for a living, to be honest.