Tibetan Buddhism did not split into competing schools the way a reader unfamiliar with it might expect, with each school inventing its own saints. Kagyu, Nyingma, Sakya and Gelug painters all depict many of the same figures, Milarepa among them, because his story belongs to the wider Tibetan Buddhist tradition rather than to any one lineage exclusively. That overlap is exactly why a museum cataloguing a Milarepa thangka cannot simply assume which school commissioned it from the subject alone.
What settles the question, in this painting, is smaller and easier to miss than the central figure: the specific style of hat worn by the lamas pictured in the surrounding narrative scenes. Tibetan painting workshops used headwear as a kind of visual signature, distinct enough between schools and sub-schools that a trained eye can assign a thangka to a particular lineage from the hats alone, independent of which saint or deity occupies the center of the composition.
LACMA's own catalogue record for this circa 1500 painting does exactly that, identifying the headwear worn by its lamas as specific to the Drukpa subsect of the Kagyupa order rather than to Kagyu more broadly, and to Western Tibet and Ladakh rather than to central Tibet. It is a small detail carrying a precise attribution, the kind of evidence connoisseurship of this tradition depends on far more than any inscription ever could.