it's a thermistor of sorts that will not function while hot
Yep, there is a PTC sitting in line with the coil and most likely connected to 110/220V.
I don't remember it exactly, but on most older TV's this circuit was always closed as soon as you switched the TV on - the PTC was kept hot due to the closed circuit, so it took about 10-20 minutes before it could fire again.
My thought is to trigger a coil with a PTC in between -don't know if this would be protection enough, but it should be the same as if you would turn your TV off and on in a short distance.
If the PTC isn't held hot all the time, degaussing should be possible after 10 minutes normal operation - no need to power the monitor down! I can't test this though for it'll take some times to get the monitor I want plus I don't know if I can hack something without voiding the warranty. Depends on if the degaussion-coil is easy pluggable and I can plug it on a circuit of my own.