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Approximately 15.9% of people living with dementia experience co-occurring major depressive disorder. Both disorders cause similar early clinical symptoms in older people but treatment options and patient outcomes differ. While it is challenging, it is therefore critical for clinicians to be able to distinguish between them. We build on existing research into objective markers of depression in speech, testing their generalizability to a more complex population. On a novel, comorbidity dataset, we demonstrate that existing depression classification methods perform worse for participants with dementia than they do for those with no cognitive decline. We also propose a method of applying Wasserstein distance-based weight vectors to emphasize depression-related information which is robust against the effect of dementia. This improves performance for users with dementia, without requiring changes to the model architectures. Our best performing model achieves an overall F1-score of 81.0%.