Feat: Add production-grade multi-stage container image for API

Two-stage build (uv builder + python:3.12-slim runtime) with non-root
user (UID 1001), no dev deps, layer-cache-optimised dep install, and
graceful SIGTERM shutdown. Verified by api/tests/build/verify_production_image.sh
covering build, health endpoint, non-root, stdout logging, secret-free
layers, missing-env-var exit, and dep-layer cache hit. All 102 integration
tests still pass; shellcheck clean.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

Makefile

@@ -1,7 +1,13 @@
-.PHONY: test-unit test-integration
+.PHONY: test-unit test-integration build-prod verify-prod
 
 test-unit:
 	cd api && python -m pytest tests/unit/ -v
 
 test-integration:
 	docker compose -f docker-compose.test.yml run --rm api-test
+
+build-prod:
+	docker build -f api/Dockerfile.prod api/ -t reactbin-api-prod:latest
+
+verify-prod:
+	bash api/tests/build/verify_production_image.sh