Can anyone recommend a source for the Tibetan army after 1354, when Tibet regained its independence from the Yuan Dynasty?
Nothing very good. There's Michael Fredholm's paper The Impact of Manchu Institutions on Tibetan Military Reform (http://www.erg.su.se/polopoly_fs/1.38956.1320402933!/Fredholm.pdf) in so far as it discusses pre-Manchu systems - which is not greatly - and a lot of the stuff in LaRocca's museum caatalogue (http://www.metmuseum.org/art/metpublications/Warriors_of_the_Himalayas_Rediscovering_the_Arms_and_Armor_of_Tibet) may relate to the period - and see also https://www.academia.edu/8493165/An_approach_to_the_study_of_arms_and_armour_from_Tibet.
I can't help noting one of Fredholm's citations: Heath, Armies of the Nineteenth Century: Asia 1, 154
While dubiously relevant to our period, I did find Fredholm's article interesting.
I feared as much.
I can't find much on the 1240 Mongol invasion either...