Well Danny, banks aren't as human as you give them credit for! What I mean is, that their process may be complex and complicated, but the goal for them is simple; the bottom line. It's not such an emotional process for them as it will be for you.
So, my answer in brief is: YES you ALWAYS want a CMA when making an offer on ANY property, bank-owned or not! It is a very important thing, but not for the bank's benefit; for yours! The bank won't care about your CMA, but you should.
As far as swaying the bank, producing things rarely sways them. What you will want to do is keep your offer and it's contingencies AS SIMPLE AS POSSIBLE. The cleaner the offer, the higher your chances. Beyond that, you just have to make it make sense to the bean counters. There's no trick to this, it's just the dollars and sense of it to them. Offer the dollars, show why it makes sense to them and, if they agree, you've bought yourself a house!