as loz says, to drain the cooler fully you`ll have to take it off unless all the ancillaries and cooler are above the thermostatic plate.
i`m pretty sure this has been asked before and a chap who worked on evo`s said he never bothered changing it in the cooler.
you`ll probably find as well that the thermo plate is ever so slightly open all the time anyway to allow a small amount of oil to circulate