Does the Stop Clock make a clicking noise (could be extremely quiet) when it turns on and off or is it completely silent? If it clicks, it has a mechanical relay and it will be fine. If it's silent, it's probably a triac.
I'm fairly sure it's a relay. In a thread on another forum, I raised the issue of using a solid state relay module for a timer and Richard of RH Designs stated that he uses mechanical relays.
It was a few years ago and I don't know if that is the case with all of his products but I suspect that it is.