fiduciary means they have a legal obligation to represent their client and no other person in the transaction in order to get their client the best deal possible. An example would be if I were listing a property and I told a potential buyer that the seller would take less, or is going through a divorce, that would be divulging personal information not in my clients benifit and therefore would be a breach of my fiduciary to that client.
As for making them lean towards you. The realtor has no decision making or bearing on who the bank sells the property to. The bank is looking at the numbers only and has no emotional attachment whatsoever. My recommendation to make your offer strong would be to make an offer that is as simple and as quick as can be. If you can close in two weeks, great, put that in your offer. That's two weeks less that the bank will have this on their books and two weeks less of carrying costs for them. Remember, for them it's all numbers and no emotion.