if I remove the led, and still measure at the two points marked, I measure 15V when Arduino I/O high, and 0V when low
Which is as it should be.
I think we need to know more about the 15V LED. No LED actually takes 15V, that is just physics, so this must be something like a chain of LEDs with built in resistors or constant current drives.
There is clearly something going on with this load that we haven't been told yet. Perhaps something to do with capacitors in the 'LED' load.