I don't think there is a good solution to this, unfortunately. Basically, if you have income that falls in the passive basket, and on which no (or little) UK tax has been paid, and you don't have any passive FTC carried forward from earlier years, and you don't have itemised deductions/standard deductions to use against the income, then US tax will be due.
With the Personal Savings Allowance coming into play, I think we'll be seeing quite a few more cases where US taxpayers such as yourself will have a small US tax liability each year, related to their UK investment income. Looking on the brighter side, the tax would previously have been payable in the UK at 40% or 45%, but now the tax will be due in the US at a slightly lower rate!