Regular language that discriminates between two deterministic CFGs
Eryk Kopczyński[1] showed in 2015 that separability (that's the name of your problem) of visibly pushdown languages by regular languages is undecidable.  The class of visibly pushdown languages is a strict subset of deterministic CFL. 

[1]:    Eryk Kopczyński, Invisible Pushdown Languages, LICS'16, available at https://arxiv.org/abs/1511.00289