I feel like this fallacy should have a name.
Here is the toy example. Alice and Bob have one loaf of bread between them. For some reason only one of them is allowed eat the bread; they cannot share it. Alice says "I should have the bread because if I don't have it I'll get very hungry and starve".
What she says is true. However it is a universal fact that applies equally to Bob and thus doesn't support her position.
Here is a more real example. Suppose there is a minority government and the Prime Minister wants to make a deal with the opposition. The PM wants the opposition to agree to support them on their next 3 years of budgets. In return the opposition gets some benefits which are not important for the purposes of this example. The PM says "We need to make this arrangement because it's impossible for the government to make any progress if they are forever at risk of being kicked out of power."
Here the PM takes the universal fact that 'A minority government is always at risk of being kicked out' and pretends it is particular evidence for their conclusion.
In other words the PM says "We need an arrangement because X". But if X was true then it would be impossible for ANY minority party to make any progress without an arrangement, and this is untrue.
Edit: I am not looking for a way to reformulate this fallacy to be an example of another named fallacy. I am asking whether the above has a specific name.