How to express a risk of breaking a target/objective

The first phrase that occurs to me for the person's emotional state is internal conflict.

The overall situation might be called a test of willpower.

You might call it the moment of truth:

a moment when a person or thing is put to the test

I'd call such a moment inner turmoil. Instead of turmoil, you could also use struggle or conflict. I also like chaos's internal.

I'm not sure turmoil or willpower or internal apply very well to the general case. This statement equally fits the question: "This bridge won't be open in time for Bastille day if we don't appropriate more funds by close of business today".

Today, in the above example, is do or die. It's make or break. It's a deadline and the engineering is giving the committee an ultimatum. All of these phrases apply to the situation. I'm not sure what the impact of your phrase is intended to have, but I think make or break fits very well, though it's colloquial.