Hi Dave,
There are two things that you may want to try.
Find a way to band-pass or “restrict” the audio going through your amplifier.
This will make your voice sound like it is coming over the radio (like a telephone voice sound).

I have never made white noise with a PIC.

The PICBASIC Pro manual says it is possible with the sound command.
See PBP manual: 5.78. SOUND
SOUND Generates tone and/or white noise on the specified Pin.

You may have to make a filter similar to the one in the DTMFOUT section.
See PBP manual: 5.17. DTMFOUT
Maybe leaving the signal as a white noise square wave would enhance the “radio” sound. This will be a low level volume. It will have to be injected into the amplifier at a low level stage. Hope someone that has used the white noise jumps in here.

If you are set-up to breadboard the PIC SOUND command, you should try it to see (or hear) for yourself. It should sound like the burst of broken squelch a radio makes when turning back to receive.

This sounds like a do-able project but it may take you a little more time that you might expect. The fun of doing it would be the reason, not the money saved.

-Adam-