Is there a way to stretch only a portion of the video?
Yes, you need to work with two video tracks, use the crop and scale effect.

Create a sequence from your clip.
Apply the crop effect and set it to left: 50%
Locate the scale settings under the always present motion effect
Disable uniform scaling and set scale width to whatever you like
Add the clip again as a new track on top
Apply the crop effect on this track too and set it to right: 50%

And that's it.