Hi
You can't stabilize a part of a clip without doing the whole clip, but a step towards what you want is:
Put your clip onto the sequence line, edit it in with any other clips that make up your video (you don't have to do this on just the clip) In this way you will probably have trimmed down the clip you want to stabilise with the rest of your video.
Either way, delete any bits not required and close any gaps.
Now place the whole time line (or bit you do want if you are working on just the clip) into the sequence bin by clicking the Plus sign in the prompt Sequence 1 X +" This will clear your timeline line and place it in the media list Sequence bin.
Open the Video media list and right click the original clip and stabilize it.
When complete follow the prompt to replace the original clip with the stabilized version.
Return to the media list Sequence bin and add your saved sequence to the empty timeline.
Your edited section will now be stabilized but the rest of the video will not be.
Note that anything saved as a sequence will behave as a single clip now once back on the timeline.
Nat