How can I modify this working circuit to reach 1V peak to peak voltage?
The op-amp you've chosen has a limited output swing. Data sheet shows a 10K ohm load (you're loading the output much more, so the output swing will be smaller). This op-amp is likely limited by its rather large voltage overhead on its output voltage swing.

 Try substituting a low-voltage rail-to-rail op-amp that can drive low-resistance loads. It must be able to supply at least 2 mA.