Also in 1813, on 15 April part of Lauriston's V Corps; Lagrange's 18th division; Suden's 2nd Brigade – 1st battalion 24 officers 600 other ranks; 2nd battalion 21 officers & 604 other ranks. They were brigaded with1st 2nd & 3rd battalions 155th regiment de ligne.
By the time of Bautzen,the 1st battalion had 25 officers and 538 other ranks, the 2nd battalion 20 officers and 434 other ranks.
By 15 August the battalions now had 1st 20 officers & 218 other ranks & 2nd 2 officers (?is this a misprint in Nafziger's Napoleon at Dresden??)& 248 other ranks. They were still in V Corps, but now in Puthod's 17 Division; Brigade Penne along with 2 battalions 134th regiment de ligne and 3 battalions 146th regiment de ligne.
The division was destroyed at the Battle of Lowenberg, 29 August. Only Major Ware, 9 officers and 30 men managed to swim across the swollen river Bober – but importantly they still had the regimental eagle. The survivors were combined with the incomplete 3rd & 4th battalions into a single battalion and by 19 December had 85 officers and 396 men. They were assigned to garrison Antwerp (Anvers)and took part in its defence during the 1814 campaign.
See Memoirs of Miles Byrne Volume 2 at link Byrne became a chef de battalion.
Guy Dempsey's "Napoleon's Mercenary & foreign Troops" has a good summary of Legion Irlandaise; but the only other major work (apart from Byrne) is John G, Gallaher – Napoleon's Irish Legion.
you may find this brief study by N. Dunne-Lynch "The Irish Legion of Napoleon, 1803-15" of use at link
likes his specialists!