I think until you better define what you're looking for, this 'fatigue' will be pretty hard to find in a convincing manner, particularly with a war two hundred years ago, different conditions, and a different psychology.
Ron's mention of the willingness of men to volunteer for a 'forlorn hope' is a good example. Would a battle-fatigued man or whole unit refuse to volunteer? I'm not sure. Such forlorn hopes were seen as a major opportunity to win recognition and advance oneself. A prime motivator for the British soldier, particularly the officers, was 'glory.' Winning recognition was a primary goal in becoming an officer. The author of "The Military Mentor" 1806 is supposedly a general giving advise to his son, now an officer. There is a whole chapter on "Love of Country" where the qualities of England are never mentioned, simply story after story of 'self-sacrifice and glory." He ends with:
“ . . . . If the principles which I have impressed upon your youth, cannot inspire you with this generous emulation, without which the path to glory and to honour, is a path only of difficulty, --stop here, and unite yourself at once with the humbler rank of citizen; I should feel less mortified by your obscurity, and it would be less disgraceful to yourself, to live unknown and undistinguished, than to fail in a career upon which you ought not to enter without the resolution to arrive at honourable distinction, or to die crowned with the applauses of your sovereign and your country.”
"honorable distintion" was very important, and for the most part not prime a motivator for today's officers, at least not in the same way with the same kind of recognition British officers hoped for. "The path to Glory and Honor" was a concrete thing for the British officer and there were identified methods for achieving it--exact moments like the 'forlorn hope' that I would imagine cut across the notion of 'battle fatigue' in a way you don't see in today's combat experience.
Of course, the actual experience of campaign was different, beside the different expectations. Captain Kincaid describes the retreat in November 13th-19th 1812, describing the infantry as 'fainting on the march', but once they reached quarters, fell out with a gusto to make the place comfortable and commenced to gamble almost immediately. Kincaid himself says he had to cut off his boots because his feet were so swollen.
Fatigue? I would imagine, but this is from a trial of a week's duration, compared to WWII infantry that were in the line for months at a time. Then, units like 95th did have a great deal of turnover. They were recruiting Spaniards in 1813-1814 in an attempt to keep up the regiment's strength. Then again, I would think that skirmishers would be 'in the line' longer than line infantry, simply because of their screening responsibilities, even between battles.
With so many variables and unique circumstances, a historian has to be very clear about what they are looking for before they have any chance of finding it. That is a basic methodology in historiography. Any set of definitions will be arbitrary to some extent, particularly about something as vague as 'fatigue'. However, if you have exact definitions, what you discover can be very concrete and enlightening history.
Best Regards and Happy Holidays.