Hi Brian,
Have you:
Placed the 4:3 video in the time-line and then right clicked on it and chosen "Set Aspect Ratio" to alter the videos perspective? There are drawbacks to this and you have to decide which and what you'd like to do.
Have you basically dealt with it as a 4:3 and then brought it into PiP - editing in 4:3 and rendering to a finished file.
The latter method would allow you to draw out the size to keep oerspective and allow you to change/select a variety of backgrounds dropped into the time-line.
Hope this gives you a few ideas.
Dafydd