Add packages:write permission to API workflow

GitHub Actions requires explicit packages:write permission to push
Docker images to ghcr.io.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

Author: koki-develop

.github/workflows/api-push.yaml

@@ -11,6 +11,9 @@ on:
 jobs:
     push_to_registry:
         runs-on: ubuntu-latest
+        permissions:
+            contents: read
+            packages: write
         name: Build and Push Docker image to Github Packages
         steps:
             - name: Check out repo