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>

infra/pgbouncer/entrypoint.sh

@@ -0,0 +1,31 @@
+#!/bin/sh
+# =============================================================================
+# PgBouncer entrypoint — render userlist from environment variables and start.
+# =============================================================================
+set -eu
+
+USERLIST_TEMPLATE="/etc/pgbouncer/userlist.txt.template"
+USERLIST="/etc/pgbouncer/userlist.txt"
+
+if [ -z "${DB_USER:-}" ] || [ -z "${DB_PASSWORD:-}" ]; then
+  echo "ERROR: DB_USER and DB_PASSWORD must be set" >&2
+  exit 1
+fi
+
+# Render userlist from template, substituting env vars
+envsubst < "$USERLIST_TEMPLATE" > "$USERLIST"
+chmod 600 "$USERLIST"
+
+echo "PgBouncer userlist rendered for user: ${DB_USER}"
+echo "Starting PgBouncer on port 6432 (pool_mode=transaction, pool_size=${PGBOUNCER_POOL_SIZE:-20})..."
+
+# Override pool settings via env vars if provided
+if [ -n "${PGBOUNCER_POOL_SIZE:-}" ]; then
+  sed -i "s/^default_pool_size = .*/default_pool_size = ${PGBOUNCER_POOL_SIZE}/" /etc/pgbouncer/pgbouncer.ini
+fi
+
+if [ -n "${PGBOUNCER_MAX_CLIENT_CONN:-}" ]; then
+  sed -i "s/^max_client_conn = .*/max_client_conn = ${PGBOUNCER_MAX_CLIENT_CONN}/" /etc/pgbouncer/pgbouncer.ini
+fi
+
+exec pgbouncer /etc/pgbouncer/pgbouncer.ini