In a nutshell, the epicurean path of human happiness (eudaimonia) is one of maximizing pleasure and avoidance of pain.
Still, the connection between this and 'frugal fatigue' is kind of iffy. The logic seems to be that maximizing spending maximizes pleasure, even though spending today predictably causes future pain. Far better to find joy in things that cost less, or endure minor discomfort now to avoid greater future pain.
The basic premise of Epicureanism is that maximizing pleasure and minimizing pain is the path to human happiness/flourishing/eudaimonia. The logical conclusion is to minimize needs in order to avoid future pain. Epicurus himself has stated that being free of pain and fear is indeed the goal.
Frugality thus becomes a strategy for avoiding future pain.
The problem is that pleasure and freedom of pain remain the focus of the epicurean practitioner who is trapped in a situation where she has to forgo a perceived good (pleasure), a minor pain, in order to avoid future greater pain.
This is the spiral of frugal fatigue as there is no end to the conundrum. The practitioner has to constantly evaluate how the present pain stacks up to the future pain avoided. But there is no way to evaluate this objectively as the future is unknown. The other problem is that fear needs to be managed.
Within Epicureanism there is no other recourse than to minimize needs to the extreme in order to inoculate against future material misfortune. Again, there is no end to this - frugal fatigue is virtually guaranteed.
The consequence of the insoluble practical problems posed, by believing that the satisfaction of desires to the greatest possible extent is the path to human happiness, is that the practitioner simply gives up. What is the point, after all, of a strategy, which taken to the logical extreme, leads to severe voluntary deprivation? And all that in the name of maximizing pleasure?
The failure mode of the epicurean is to break the frugality trap by seeking satisfaction in the present. That often results in substantial credit card bills.
The stoic, on the other hand, questions the viability of the proposal that satisfying desires, even properly managed, is a path to happiness at all.
This turns into an investigation into the nature of desire. Interestingly, the epicurean excesses show up in the nature of desire itself. The attempt at satisfying desires turns out to be an infinite recurse which can only be interrupted by extinguishing desire completely. This is what the epicureans found - and typically grow tired of quickly. There appears to be no middle ground as long as satisfaction of desire is a primary objective.
There is another misconception that stoics try to eliminate desires (passions). This is not true. Stoics are students of the passions.
Here is an exercise for you (I use an ice cold coca cola, you may substitute a piece of cake or a Lamborghini): you are thirsty and you go into a store. Visualize an ice cold coca cola while you walk to the cooler. Take it out and observe the condensation on the bottle. Imagine the sensation of drinking the heavenly liquid. Put the bottle back. Go to the water fountain and quench your thirst. Leave the store.
You may have to do exercises like this several times to develop the observation skills needed to see the desire growing and subsiding without ever having satisfied it in the traditional sense.
Of course, the stoic does not believe that desires can actually be satisfied in a traditional sense. Desires arise and dissipate. Desires are ephemeral and their dissipation is their satisfaction. The object of a desire is in the last consequence immaterial to its satisfaction.
The consequences of separating desires from their object as seeing them as mental events independent of their apparent object are profound: the world instantly ceases to be some sort of big department store simply there to satisfy ones desires.
Stoics also see the attempt to satisfy a desire by obtaining its object as a futile attempt at extinguishing the desire for good. This is of course not possible as another desire will take its place and so on. Stoics accept desires as part of the human condition and watch the parade with equanimity without letting it interfere with what really matters in life.
So here you have it: Stoics do not believe that one stays any chance of actually satisfying a desire with an external object and thereby getting rid of the desire for good. Desires come and go and acting upon them or not is entirely up to us.
In the stoic view, frugal fatigue does not exist as it is based upon a misconception of the nature of desire. Or, in other words, frugal fatigue is the inevitable consequence of the belief that desires beyond the immediate bodily needs of an organism can be satisfied by external objects. The stoic is convinced that the satisfaction of a desire is immanent in its nature as a transient mental event and is manifest in its dissipation. And dissipate it will - every time, and that's why the stoic life is a life of plenty.
Peter
Edited for typo