It seems that the Romans typically called their sun-god by the name of Deus Sol, not Sol Deus. Therefore I would not expect the Christian use of 'Soli Deo' to have been confusing or ambiguous to them.
According to Manfred Clauss in The Roman Cult of Mithras (1990,
translated 2001), chapter 13, Mithraists referred to their
sun-god as Deus Sol (not Sol Deus)....