Sure you can use an interposing type relay, though for the load you listed the STC-1000 will probably switch it just fine.
Only thing in that drawing above would be to label the neutral and hot. I'm assuming the far left wire is neutral. Then to the 120V coil relay, maybe break the hot instead of the neutral. Breaking the neutral usually isn't done even in control circuitry, with the exception of motor overload heater contacts.
The other thing you could do there is get a 12VDC STC-1000 and a 12VDC interposing relay; main benefit there being then you have only low voltage at your STC-1000 though it really doesn't make much difference.