The 50% figure is often brought up to compare the large role/importance of the Roman military in comparison to the proportionally smaller investments made in the armed forces by modern countries - but wouldn't Rome, an incredibly advanced bureaucratic state for the time, have had more non-military (infrastructure, etc) financial obligations than other city-states/empires of the time? It seems like the 50% figure should really be on the low end of military expenditures, in the context of the era.
I might want to mention that whilst I can't confirm or deny the Southern Song figure (is there a source?), Southern Song existed in almost a state of perpetual total defensive war against the Jurchens or Mongols.
If you look at a more "normal" period of the Chinese Empire like the Eastern Han Dynasty (which granted is also an incredible avanced bureaucratic state of the time)....it's very hard to calculate a percentage but the Eastern Han Empire was estimated to spend relatively, very little on its standing military (which comprised of only border garrisons and a professional army of 4,000 in the capital). Scheidel estimated that for the Han Empire's military expenditure to match Rome's, it would need a standing professional army of half a million, which basically never came close to happening.
Source: State Power in Ancient China and Rome, Walter Scheidel