There's no type of relay (at least that I know of) that does this very specific task.
You're trying to solve an easy control problem with a complex electromechanical setup. Since this is error-prone, relatively expensive and complicated, we don't do that since the late 1970s, when microcontroller became ubiquitous.
So, use your existing standard relay, buy a < 1€ microcontroller, take a minimal example from the vendor, change it to toggle a pin after the times you need, and connect a transistor to switch your relay to said pin.
If you're hesitant about the complexity of a microcontroller design, buy the cheapest microcontroller eval board you can find: these come with all the electronics you need to supply power etc to your microcontroller, a programmer built in, and a wealth of examples. And although the microcontrollers on these boards are typically 32 bit powerhorses compared to what you need, they're < 15€ for the full board. In exchange, you got so plenty of resources you effectively don't have to worry about programming efficiently, and get interesting extra stuff like watchdog timers (which can be used to reset everything if something hangs), ADCs (which can be used e.g. to monitor motor currents) and enough pins to connect displays and user buttons.