You could try to use a 74HC5555, it is a 555 with a internal divider for long time periods. If you set it to 17 bit (131,072) and use a 1 KHz oscillator you would get 2.184 hours, 1.1 KHz would be 1.96 hours.
You would use the Q output to drive Bradtherad's decade counter and use the 74HC5555 non-retriggerable mode. Meaning that the 2 hour period could not be extended every time you hit the go button.
The problem i see is that you really need a astable multivibrator not a monostable. After the first 2 hour timeout the monostable will sit there waiting for another input to tell it to start another 2 hour timing session and advance the 4017 counter.
That's the problem with these "simple" circuits. They snowball quickly in complexity. The quick dirty fix is feed the monostable with a regular 555 astable circuit running say every 30 seconds. So the monostable running in non-retriggerable mode will ignore the multiple input pulses until it is done timing and then it will retrigger for another 2 hours.
Of coarse there are other practical matters like the power switching with the relays that must be addressed and a power supply and a box etc.
This is a small project in itself and some electronic design knowledge is necessary.
The best answer may just be get a low cost programmable logic controller as mentioned if you can find one with at least 3A output relays. 300W/220 vac = 1.36A, and for a long life time use 3A or higher rating.