feat(infra): add PgBouncer connection pooling for production PostgreSQL

Introduces PgBouncer as a connection pooler between the API service and
PostgreSQL in docker-compose.prod.yml, reducing connection overhead and
improving concurrency under production load.

- Add PgBouncer service (edoburu/pgbouncer:1.23.1-p2) with transaction
  pool mode, max_client_conn=200, default_pool_size=20
- Route API DATABASE_URL through PgBouncer (port 6432), keep direct
  connection (DATABASE_URL_DIRECT) for Prisma migrations/introspection
- Create infra/pgbouncer/ config: pgbouncer.ini, userlist template,
  and entrypoint script with runtime env-var substitution
- Update prisma.config.ts to prefer DATABASE_URL_DIRECT for migrations
- Add K6 load test (e2e/load/pgbouncer-pool-test.js) with ramp-up to
  200 VUs, pool exhaustion detection, and p95 < 2s threshold
- Add PgBouncer env vars to .env.example

Co-Authored-By: Paperclip <noreply@paperclip.ing>

docker-compose.prod.yml

@@ -12,7 +12,9 @@ services:
       - '${API_PORT:-3001}:3001'
     environment:
       NODE_ENV: production
-      DATABASE_URL: postgresql://${DB_USER}:${DB_PASSWORD}@postgres:5432/${DB_NAME}
+      DATABASE_URL: postgresql://${DB_USER}:${DB_PASSWORD}@pgbouncer:6432/${DB_NAME}
+      # Direct connection for migrations (bypasses PgBouncer — required for DDL)
+      DATABASE_URL_DIRECT: postgresql://${DB_USER}:${DB_PASSWORD}@postgres:5432/${DB_NAME}
       REDIS_URL: redis://:${REDIS_PASSWORD}@redis:6379
       TYPESENSE_HOST: typesense
       TYPESENSE_PORT: 8108
@@ -27,7 +29,7 @@ services:
       AI_SERVICES_API_KEY: ${AI_API_KEY}
       RUN_MIGRATIONS: ${RUN_MIGRATIONS:-false}
     depends_on:
-      postgres:
+      pgbouncer:
         condition: service_healthy
       redis:
         condition: service_healthy
@@ -165,6 +167,49 @@ services:
     networks:
       - goodgo-net
 
+  # ── Connection Pooling ─────────────────────────────────────────────────────
+  pgbouncer:
+    image: edoburu/pgbouncer:1.23.1-p2
+    container_name: goodgo-pgbouncer
+    restart: unless-stopped
+    entrypoint: ['/bin/sh', '/etc/pgbouncer/entrypoint.sh']
+    environment:
+      DB_USER: ${DB_USER}
+      DB_PASSWORD: ${DB_PASSWORD}
+      PGBOUNCER_POOL_SIZE: ${PGBOUNCER_POOL_SIZE:-20}
+      PGBOUNCER_MAX_CLIENT_CONN: ${PGBOUNCER_MAX_CLIENT_CONN:-200}
+      PGBOUNCER_ADMIN_PASSWORD: ${PGBOUNCER_ADMIN_PASSWORD:-pgbouncer_admin_secret}
+      PGBOUNCER_STATS_PASSWORD: ${PGBOUNCER_STATS_PASSWORD:-pgbouncer_stats_secret}
+    volumes:
+      - ./infra/pgbouncer/pgbouncer.ini:/etc/pgbouncer/pgbouncer.ini:ro
+      - ./infra/pgbouncer/userlist.txt.template:/etc/pgbouncer/userlist.txt.template:ro
+      - ./infra/pgbouncer/entrypoint.sh:/etc/pgbouncer/entrypoint.sh:ro
+    depends_on:
+      postgres:
+        condition: service_healthy
+    healthcheck:
+      test: ['CMD-SHELL', 'pg_isready -h 127.0.0.1 -p 6432 -U ${DB_USER}']
+      interval: 10s
+      timeout: 5s
+      retries: 5
+      start_period: 10s
+    deploy:
+      resources:
+        limits:
+          memory: 256m
+          cpus: '0.5'
+        reservations:
+          memory: 64m
+    security_opt:
+      - no-new-privileges:true
+    logging:
+      driver: json-file
+      options:
+        max-size: '5m'
+        max-file: '3'
+    networks:
+      - goodgo-net
+
   redis:
     image: redis:7-alpine
     container_name: goodgo-redis