You should be able to avoid removing the bar tape by:
1. Unbolting and removing the shift lever from the shifter body.
2. Pulling the broken cable pieces out of the lever and housing from both ends.
3. Rethreading a new cable into the lever and push it completely through the housing
4. Pull it snug and reattaching the lever to the body.
5. Reattach the cable to the derailleur.
You will need a brand new cable with the derailleur end intact so it will thread through the housing without snagging.