Sounds like quite a challenge. I have never attempted such a repair but...
I would consider this - You can get medical needles from your local Pharmacy.
Next, I would mix a small batch of epoxy and warm it to reduce its viscosity and
test the ability for the needle to push the glue. I believe you can get needles with
different diameters. If this proves to be acceptable, you may be able to drill some small holes
in the cork and inject epoxy. Leave room at the top of each hole for some cork dust/white glue
mix. Sand smooth. I would use a pen style pin vice to make the holes very carefully using a 1/16
drill bit or smaller. The issue would be drilling without damaging the blank under the cork. Being a Winston
I would rebuild the grip and reel seat as a proper repair...
And, if the epoxy will not push through the needle, you may be destined to rebuild.