If I just add an 10mV sine input instead of the 300mV that I am getting from the oscillator, the circuit works properly. Can I reduce the voltage in the oscillator so that the output doesn't saturate or is there any way to change my diff amp in order for the output signal to not saturate.
Multiplying oscillator signal with a pulse
Your thinking seems muddled. Because: -
My output when I feed a very small input sine wave into the input of the diff amp and my required output
The so-called required output is not multiplication - it is generally a straightforward addition. But, it's unclear why you want to add two signals (as per the 2nd circuit) when your title is about multiplying. Your top circuit multiplies but your bottom circuit just adds. Be clear and explain what you are ultimately wanting.
If you want multiplication then use the differential output from the two collectors in the top circuit - that gives you amplitude modulation aka multiplication. In case your unsure about what I'm saying: -