Why on earth were you fiddling with the fault test switch when the warning illuminated? And why did you ignore an engine overheat? An overheat warning is for overheat of the engine shroud, not the engine core, so the engine indications are not likely to be relevant - overheats are almost invariably hot gas leaks into the shroud or nacelle, which as mentioned already, can cause all sorts of problems with fuel, electrical and hydraulic systems, not least fires.
I would suggest that ignoring the warning and maintaining climb thrust until the cruise shows a serious lack of understanding of your aircraft and procedures. Best get back into the books. The fact that the warning stopped as you throttled back for the cruise shows that this was the case and that you should have followed the QRH (and stopped the climb).