Despite countless attempts, sice the dawn of Man, to make man conform to some perceived 'need' or 'emergency,' the data is clear: Vary from the "norm" and you WILL pay a price.
The French Revolutionaries experimented with 10 hour days and 10 day weeks.
The Soviets began beating back the German invaders right about the time they admitted that rest was necessary. Coincidence? Well, they certainly thought it was proven well enogh to justify shooting commanders who failed to allow for adequate rest. Quite a change from the retreat, when they shot commanders for the opposite reason.
The fact is, while it is well established that 'fatigue' causes bad things to happen, it's devilishly difficult to either define or measure fatigue. It's also amazingly difficult to determine exactly what increases ior lessens fatigue.
Probably the most damaging effect of fatigue is that your judgement becomes unreliable. Forget about falling asleep at the wheel; instead, it makes perfect sense to drive hundreds of miles in the wrong direction.
One really can't discuss, or address, the issue without challenging the very foundation of 'industrial engineering,' as expounded by Taylor. In his model, there's absolutely no room for the individual; it's all clipboards and stop watches. "Fatigue," as a concept, simply does not exist.
Unfortunately, it is the "Taylor" framework that drives Union contracts and work rules; the rigid system does not allow for a man to take breaks as needed, but only at defined times. About all you can do is play the 'safety' card; this is hard to do successfully, as there is invariably someone else available who is not fatigued. Thus, you wind up trying to explain "I can do it now but not then."
So, one is certain to have to make a stand at some point. Take the break and take the consequences. Since they've made it clear that they don't need you, let them have it their way- without you.