How can I add a GFCI breaker on two circuits with a shared neutral without rewiring?
If you are asking if you can double tap the breaker in the panel (#1), the answer is no.
GFCI Breaker
My first thought in this situation, would be to use a 20 Amp double pole GFCI breaker.

These tend to be quite expensive, but it will protect both legs of your circuit.
Single GFCI Receptacle
The next option would be to install a single GFCI receptacle to protect all the downstream receptacles, and leave the lights unprotected.  Though you'd have to check your local codes to know if the lights need to be protected.

Just remember, you'll have to pigtail the neutral before the GFCI receptacle for this to work.
Two GFCI receptacles
The other option is to install a GFCI receptacle to protect downstream receptacles, then install another GFCI receptacle to protect the lights.

Again, you'll have to pigtail the neutral before the first GFCI receptacle.  This also has the disadvantage of having a receptacle on the light circuit, which can lead to overloading the circuit and standing in the dark because of it. However, it does insure both legs of the circuit are protected.  You could install the GFCI receptacle that protects the lights in an inconvenient location (while still keeping it accessible for resets/tests), to limit the temptation to use it.
Remember to check all local codes and regulations before doing any work, and never be afraid to call an Electrician if you feel uncomfortable at all during the installation.