Assuming a 1A+ USB supply on the primary side, what is the maximum continuous current that the isolated side can draw?

Maximum continuous current that can be drawn from the 7054-based isolators is 500mA. The output voltage will drop if more than 500mA is drawn and shuts down in hiccup-mode at 800mA.
Sustained short circuit-proof: the unit was tested on short circuit for 1 hour and no damage occured.