I don't understand what Chashad there would be that the Mumcheh is "changing his ruling for the benefit of the benefactor" unless there would be a personal gain for that Mumcheh. Rashi even states what the problem is; it's because of the agra, the reward or wages involved, that the Mumcheh will come to permit a 'tam' animal to be slaughtered. Why then can't any Mumcheh be paid provided that he's paid once and regardless of his psak?
Maybe the Mishnah is ignoring the fact that a Mumcheh is always paid. Before the Rabbanan would extend the owner's obligation to pay from the case of ba'al mum to a case of tam, they would decide who was worthy of this takana (to receive payment altogether) in the first place. A Mumcheh met their standards if he would not be nechshad in the scenario that he only receives money for a beneficial psak. Once the Mumcheh met these standards, the Rabbanan allowed him to make money, but told him to collect regardless of the psak in order to prevent the chashad of the masses. This is how I understood your original answer, but it doesn't seem to fit with the chashad of 'benefiting the benefactor,' so I wanted to ask if I'm still misunderstanding this mishnah.
Thank you for your patience and effort in helping me understand this sugya,
I think that the answer lies in one word: competition. If every Mumcheh would be paid once, regardless of his Psak, you would have a scenario where Mumcheh A is honest, but requires people to come back until the animal is actually a Ba'al Mum, versus Mumcheh B who for some reason is rarely known to require people to come back for a second check, as he always says the animals are Ba'alei Mumin. Who are people going to go to? B's practice of "benefitting the benefactor" clearly keeps the business coming his way. Wanting to avoid such a scenario, the Rabanan made strict rules about who can see Bechoros.