If I recall rightly, Duke's wording implies that he will pay for the bounty, but in reality you only get the normal money reward for completing a mission.
I believe the question is how much you can place on the bounty required for that mission without losing money. This would stem from some confusion surrounding the wording of the mission itself.
You won't be refunded any money related to the bounty. Instead, the "refund" is really just Duke's way of saying that your payment for the mission should be enough to cover the required bounty in order to make sure your target gets hit before the mission expires. The money listed when you accept the mission is all the money you'll receive from it, and you'll receive that amount no matter. Therefore, the less money you place on the bounty, the more you'll make once the mission is finished. Note, however, that you still need the bounty to be claimed, meaning that you either have to have a friend willing to make the attack for almost nothing, or place a high enough bounty that it gets claimed (I've found that 100k is usually sufficient, and is significantly less than the payment from the mission).