I believe the WARN light should come on if the coolant temp sensor is bad.. chances are you have a failed solder joint on the circuit board of your gauge cluster its vdery common..
the 444E temp gauge gets is signal from the computer.. the 5 main gauges(tenp, oil P, speed, RPM, Volts) are run on a data link from the computer.. I believe the warning bell is also just a feed from the computer, id have to look that up in the book.. I do know a shorted or open sensor will issue a code, you can check your codes wit hthe diagnostic button under the dash.
you alarm comes on at coolant temp 230....
you can fix it how you like.. im not the one to ask because I almost always repair bad gauges the way the facotry intended even if it costs more..