In the sense in which the minimal length is present is standard string theory, in the same sense the minimal length is present also in Bohmian string theory. However, Bohmian mechanics claims that standard quantum theory is incomplete, i.e. that one must add something additional that standard quantum theory doesn't have. So one can say that one part of the theory does have the minimal length (as required by quantum gravity, as you said), while the other part of the theory doesn't have it. This other part does not need to have the minimal length for consistency.As I understand, the minimal length is crucial to any theory of quantum gravity in order for that theory to be consistent. You can look it up yourself.