US Tax treatment of dividends from single member consulting firm, a UK Private Limited Company

US citizen in UK has a salary from one firm which includes National Insurance withholding, and also owns a UK Private Limited Company to which client payments are made for his consulting work. UK tax law allows payments to him to be just dividends, with no employment taxes. Company is a Disregarded Entity for US Tax, so dividends are not recognized – just income is recognized. Company is not a passive investment, so it seems it would need to go on 1040 Sch C, with earnings considered as earned income for US tax, notwithstanding UK characterization as dividends. Is this correct?
Result of Sch C wants to go to Sch SE, but Totalization Treaty with UK says if covered by UK Nat’l Ins, you need not pay SE Tax. Is this true, even though salary alone is well below SE upper bound? (Could a UK llc pay £10 in salary subject to PAYE, and not have to pay SE Tax on £100k in dividends from that UK llc consultancy?)