Beyond the rating, the thermocouple calibration is not guaranteed, it won't damage the metal per se, but that depends on what type of wire the thermocouple is made off before it too is damaged. The insulation on the thermocouple is rated usually + 50%, so it will melt at ~ > 300+C.
For a hot air gun rated to 500C I'd get an K type in eBay, they are fairly low tech and generally reliable. You can also test its accuracy against a water bath of ice and boiling water. I got one 800C for $12, its encased in steel mesh and has heavily metal/glass insulated wires.