Did you look at this case? https://www.amazon.com/Argon-Raspberry-Heatsink-Supports-Accessible/dp/B07WMG27T7/ref=sr_1_2?dchild=1&keywords=pi4+argon&qid=1592447696&sr=8-2
I have it and am a fan. The whole thing is a heat-sink except for the very bottom piece, which I was a little confused why they made that part plastic. The lid would probably slide right over the POE hat.... Probably, 99% sure just looking at the photo of the hat and holding the case.
If you have stand off's and are handy with a soldering iron you could potentially modify the fan hat attachment for the case to fit the POE hat if you really needed the cooling but the lid definitely would not fit any more
RE: Raspberry Pi 4 Heat and Enclosure Issues