Cameron Norrie beat Taylor Fritz in three sets but Great Britain trail their United Cup play-off against the United States 2-1.
British number one Norrie won 6-4 5-7 6-4 in the Sydney City Final, after team-mate Katie Swan had earlier lost 2-6 6-3 6-4 to American Madison Keys.
Harriet Dart was then beaten 6-2 6-0 by world number three Jessica Pegula.
Dan Evans faces Frances Tiafoe in the final singles rubber in a bid to keep alive GB's hopes of progression.
Should he win, he and Dart will then take on Tiafoe and Pegula in the deciding doubles.
The Sydney City Final is one of three City Finals taking place across Australia, which act as a play-off for the semi-finals.
The winners in Sydney, Perth and Brisbane will progress, as will the runners-up with the best record.
GB - captained by Tim Henman - are the surprise package at the inaugural mixed team United Cup, having upset Australia and Rafael Nadal's Spain in the group stage.
Elsewhere, Jack Draper is out of the Adelaide International after being beaten in straight sets by Karen Khachanov.
The eighth seed won 6-4 6-2 and will play third seed and fellow Russian Daniil Medvedev in the quarter-finals.
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