Hi,
Can you consider using an astable 555 config. triggering another astable config rather than a monostable triggering an astable config ?
If you can do so, simply set the time period of the first astable config for the duration for which the second astable config has to be switched off. Then, connect a P transistor between supply and vdd pin of second astable config. connect the ouput of first astable config to gate/ base of p transistor.
so now, when the first astable conif switches on, it triggers the p transistor into cut off region, swithcing it off and thereby switching off the second astable config. After the set time period, the first astable config switches off, reducing the voltages of the gate, thereby switching on the p transistor and the power supply to the second astable config (switchint it on automatically)