Should be possible with Aux instead of K2.
Or you can use K1 alone, but you need an external relay and a 230V time delay with on and off delay as one function.
K1 controls the secondary relay coil.
230V genset voltage controls the time delay coil.
Time delay common is fed 24V (or whatever the genset needs).
Time delay NO goes to secondary relay NC.
Time delay NC goes to secondary relay NO.
Secondary relay common is your start/stop pulse output.