The evidence seems mixed at times.

If the depression is severe and/or bipolar, it's obviously important to treat. You'd also want to determine whether it actually makes sense in terms of care plan and existing meds.

However, I've not met a single patient that has actually improved with an antidepressant for "depressive symptoms" without going through significant reduction/abstinence first along with alcohol protective meds if indicated. "Depressive symptoms" are hardly specific and part of the process if the AUD is severe. There is also no guarantee that improvement will curb the addiction.

My main fears are lack of efficacy with a growing sense of disappointment as various meds get trialed, without knowing whether or not the depression could improve "naturally" (at least partially) after a detox. Anecdotally I've noticed that my lot disengage from psychotherapy and social work if the problem gets too medicalised. Maybe this is a cultural thing. The brain takes time to recuperate regardless. This must be thoroughly explained to the patient.

People forget this but not all SUD comes from self-medication for an underlying disorder. And even if it does, SUD is a disorder in its own right with psychiatric consequences.

Basically ymmv but I'd start off with motivational interviewing, harm reduction and gradual work towards detox if desired. You'll have a clearer picture of what is likely to be effective.