From a consequentialist perspective it doesn't matter whether she's lying or not: she could be lying, so if I'm seen to grant her an extension I create incentives for others to lie, which is the thing I as a consequentialist actually want to avoid. (Of course, in the real world there are ways I could find out, but that's outside the scope of this example.)

Reducing the size of the incentive doesn't actually address the issue. That creates less incentive, sure, but the basic problem remains unaddressed.

The more seriously we take the possibility that she's lying, the more Grieving Student starts to resemble Lazy Student or Sports Fan.