One of the difficulties with trying to identify units which might have exhibited signs of battle fatigue is defining the difference, if any, between "battle fatigue" and "battle smarts".
Does a veteran unit which falls back because its flanks are exposed or it believes that they are exposed exhibit 'battle fatigue' or 'battle smarts'? Would failing to press home an assault on a position that they recognize from experience cannot be taken 'battle fatigue' or 'battle smarts'?
It would seem that 'battle fatigue' might explain why a unit falls apart after slight pressure, but without a thorough examination of its previous battlefield experiences, length of time in combat, composition of the unit at the time [% veterans; % new recruits], make-up of its officers, discipline on the march and in cantonments, etc., the situation it was in at the time, how can one be certain?
A number on instances in the peninsular army where units fell back would seem to be more from the direction or misconduct of the officers in command. Glover noted that after reviewing the General Orders for courts-martial information, he could not find an instance of one for cowardice in the face of the enemy.
George Napier wrote that when a call for 100 volunteers for a forlorn hope at Ciudad Rodrigo was made almost half the division volunteered and "we were obliged to take them at chance." Again this was in the Light Division. At the siege of San Sebastian, Wellington being displeased with the effort of the Fifth Division called for volunteers from the First, Fourth and Light Divisions to mount another assault [see account of Colonel Hunt in Verner's History of the Rifle Brigade].
We know Wellington's feeling regarding sending home under-strength veteran units to be replaced by larger untried units. He preferred to combine them in provisional battalions. Not something he would do if these units were exhibiting signs of 'battle fatigue'.
James Alexander would have been a relatively junior officer [Captain] in 1839 and a few years before had been knighted for his services in Africa when he edited that work. He may have been on the half-pay of the Black Watch at the time. He had yet to see the fighting in the Crimea where he served as Major [Brevet Lieutenant Colonel] and commanded the 14th Foot. He would have seen the 2nd assault on the Redan where the hard used 2nd Division and Light Divisions [name was historical only] were repulsed. One of the reasons for the failure was ascribed to the fact that the regiments in both divisions were made up of recent and young recruits and not old hands. A general feeling was that the 1st Division and the Highland Division should have made the assault, both not having seen such hard service and having a higher proportion of veterans in the ranks.
There may also be a psychological element to this question. In World War II, Orde Wingate held to the theory that men could only fight in the jungle for three continuous months and then had to be rotated out. He applied this to his Chindit Columns. Now a fine Allied unit had fought for three months in the jungle and when ordered to continue to advance, the unit collapsed. The subsequent investigation found that the men had been exposed to Wingate's theory of three months and out. However, another Allied formation went on to fight successfully for considerably longer in the jungle. It was discovered that this formation had never been exposed to Wingate's theory. So at some point do units feel that they have done their bit?
Given the campaigning tactics of the period, it would seem that constant combat was not a feature, except perhaps for sieges, and so a greater opportunity to rest, train and restore a unit. So then a greater emphasis on individual 'battle fatigue' as opposed to unit 'battle fatigue'?
Just some thoughts and observations, not very scientific.