To power the Muon over PoE I need a PoE HAT and move the jumper on the board to connect the 5V_IN and center pin. All clear.
My question is:
Will the battery be charged when powered by PoE and will it switch automatically to battery power when the PoE power is lost?
The Muon will charge the battery from PoE when the jumper is set correctly. The Muon will switch to battery if PoE power is lost, as well.
The HAT 3V3 will be powered from the Muon, but HAT 5V won't be powered, since that's configured as a power input, not output, when using PoE.
Great, thank you for the clarification!
Time to order the PoE HAT 
Make sure you get one designed for the Raspberry Pi 5.
The Pi 4 PoE HAT connectors are in a different position and don't fit the Muon.
Yes I will go with the Waveshare one mentioned in the Muon datasheet.