perf: reuse a shared reqwest::Client across proxy requests

[ttatsato]

Problem

src/routes/proxy.rs:66 creates a new reqwest::Client on every proxied request:

let client = reqwest::Client::new();

reqwest::Client owns its connection pool. Constructing one per request means the pool is thrown away after a single use, so HTTP keep-alive to upstream services never takes effect and every proxied request pays a fresh TCP (and TLS, if applicable) handshake. For an API gateway this is on the hot path and shows up directly in p99 latency.

This is the most clear-cut performance issue in the proxy hot path today and is worth fixing even at the MVP stage.

Proposed fix

• Build a single reqwest::Client at startup in main.rs.
• Wrap it in Arc<reqwest::Client> (or rely on Client's internal Arc and clone it directly — Client::clone is cheap).
• Pass it through create_router and into the proxy handler via Axum State, alongside the existing PgPool.
• Use the shared client in proxy() instead of reqwest::Client::new().

Notes

• No behavior change expected for callers; this is a pure performance fix.
• Consider setting reasonable defaults on the shared client (timeouts, pool idle settings) in a follow-up — out of scope for this issue.