How can I check the power draw of an application?

Wow, thanks for that experimentation! I'd say yes, that looks like it could be pretty good.

Am I correct in assuming that the systick overflows every 262ms?

That sounds about right, yes.

Interestingly we do have a SysTick_Handler in jshardware.c which is called when SysTick overflows. I wonder if we could do this backwards - by checking the RTC in that systick we could see how much time we were asleep over those 16M ticks, without needing any extra wakeups.