One of the main features of the Flirc RPi case (besides awesome looks :)) is that it also acts as a heat sink for the CPU. There's no way to do that with the HAT, as it covers the CPU.
Some time ago some user removed the plastic cover on top of the case (it's easily detachable) and cut a hole with a dremmel right above the GPIO port and built a GPIO extension which raised the GPIO port high enough that the HAT could be placed on top of the Flirc case. Then he 3d printed the case for the HAT which attached to the top of the case (in place of the original plastic cover).
I've started doing the same but I didn't bother with making a makeshift GPIO extension but I've found a 40-pin goldpin female header with long male pins on the other end, which is more than enough to raise the port on its own.
What I would like to see is the GPIO hole already done at the factory and ability to download a 3d model of replacement cover to print out. It could then be adapted to whatever HAT someone would like to use with the case.
Here's the extender:
And how it looks in the case when mounted on the RPi 3: