Would a UK resident who pays tax to a foreign country on dividends (of less than <£2,000) need to file Self Assessment for this reason?

Suppose an American expat taxpayer living in the United Kingdom earns dividends on U.S. stocks that fall below the UK’s tax-free dividend allowance of £2,000, and this is the only income they have from outside the UK. Assume that they do not have capital gains, or any other circumstances that would cause them to have to file Self Assessment.

It seems that such an individual does NOT have to file Self Assessment.

But does the situation change if the taxpayer has to pay some nonzero amount of U.S. tax on these dividends? (I have read that generally speaking, if a person is a UK tax resident and needs to pay tax on income from outside the UK, then they have to file Self Assessment, but I’m unclear on whether that applies to this specific edge case.)