The built-in (or system) apps are not restrained in the same way that 3rd party apps are. There are many examples of this on Windows Phone but two visible ones are...
The calendar app updating its Live Tile even when Battery Saver is turned on (which you noted).
The People Hub Live Tile which has many small tiles which rotate in sequence - as developers, we don't have access to that Live Tile Template.
The Photos Hub Live Tile displays photos using a panning animation - also not available to developers.
As there is no 'fix' if your PeriodicTask
doesn't run as expected, you'll simply have to update the tile when your app runs as well as checking the status of the PeriodicTask
each time.