feat(ai-router): test an endpoint before saving it, and re-point a live lane

Two gaps left by the AI Router feature: a profile could be saved without ever
being reachable, and a lane's router could only be chosen at creation.

**POST /api/ai-routers/test** answers the question saving cannot. It sends one
Anthropic-style message capped at a single output token to `<baseUrl>/v1/messages`
with a 15s timeout — the wire shape a claude lane's eight env keys point at, so
what is tested is what will actually run. A router that answers is a 200 here
whatever it replies: `{ ok, status, message }` carries the router's own words,
and `status` is null when nothing answered at all. Only bad input is a 4xx.

The stored key is only ever sent to the endpoint it was stored against. A blank
`apiKey` falls back to the key of the profile named by `name`, but only while
the submitted `baseUrl` still equals the saved one — so testing a retyped
endpoint needs a retyped key (`EBADKEY`). Without that rule this route would be
a way to make the server POST a saved API key to any URL a browser names.

Network failures report `error.cause`, not `error.message`: node's fetch reports
every transport failure as the same useless "fetch failed" and hides the part
the user can act on (ECONNREFUSED, ENOTFOUND, a TLS error) one level down.

**The lane card carries an AI Router select.** The Add-lane modal was the only
place a router was chosen, and a lane outlives its creation. Switching goes
through `POST /api/lanes/:id/router/apply` — never a lane PATCH — because the
DB column and the lane's own config on disk may only move together. A failed
apply puts the select back rather than leaving the card claiming a router the
lane did not get. Hidden when `native` is the only profile: a select with one
option is a control that cannot do anything.

Adds `Lane.router_profile` to the client type (the payload already carried it).

docs/API.md

@@ -1723,18 +1723,23 @@ Every history row carries `lane_id` (the lane whose run it belongs to) and `tool
 ```
 GET    /api/ai-routers                Every saved profile, the built-in `native` one first
 POST   /api/ai-routers                Create — Body: { label, baseUrl, apiKey?, model, models? }
+POST   /api/ai-routers/test           Probe an endpoint — Body: { baseUrl, model, apiKey?, name? }
 PATCH  /api/ai-routers/:name          Update the same fields, all optional
 DELETE /api/ai-routers/:name          Delete one saved profile
 ```
 
 `PATCH` and `DELETE` also answer with `lanes: { updated, failed[] }` — see "an edit reaches the lanes" below.
 
-A **router profile** is an endpoint that stands in front of Claude Code or Codex (9Router, OpenRouter, a self-hosted gateway) plus the model names to use against it. It powers the **AI Router** tab of Settings (`/settings?tool=ai-router`), which stores profiles and attaches nothing; attaching happens once, when a lane is created.
+A **router profile** is an endpoint that stands in front of Claude Code or Codex (9Router, OpenRouter, a self-hosted gateway) plus the model names to use against it. It powers the **AI Router** tab of Settings (`/settings?tool=ai-router`), which stores profiles and attaches nothing — the tab's **Test** button (`POST /api/ai-routers/test`, below) is its only outbound call. Attaching a profile to a lane happens in the "+ Add lane" modal, or later from the lane's card in Workspace.
 
 `models` carries the five per-tier names (`opus`, `sonnet`, `haiku`, `smallFast`, `subagent`). Any tier omitted is filled from `model`, so a router that answers to one symbolic name needs one field, not six.
 
 **No response here ever contains key material.** A saved profile reports `hasKey: true|false` and there is deliberately no `GET /:name` — a route returning one profile in full would be the single way an API key could leave the server. Profiles live one JSON file each under `<CLAUDE_HOME>/agenthub-routers/` (dir `0700`, files `0600`). An omitted `apiKey` on `PATCH` keeps the stored one; the edit form cannot display a key, so it has none to send back.
 
+**`POST /api/ai-routers/test`** answers the question saving cannot: does this endpoint, with this key, actually serve this model? It sends one Anthropic-style message capped at a single output token to `<baseUrl>/v1/messages` (a 15 s timeout), which is the wire shape a claude lane's eight env keys point at — a Codex-only router is not covered. A router that answers is a `200` here whatever it replies: `{ ok, status, message }` carries the router's own words (truncated to 300 characters), and `status` is `null` when nothing answered at all. Only bad input is a 4xx.
+
+The stored key is only ever sent to the endpoint it was stored against. A blank `apiKey` falls back to the key of the profile named by `name`, but **only** while the submitted `baseUrl` still equals the saved one — so testing a retyped endpoint needs a retyped key (`400 EBADKEY`). Without that rule this route would be a way to make the server POST a saved API key to any URL a browser names.
+
 `native` is a built-in profile, not the absence of one: empty `baseUrl`/`apiKey` and the vendor's own model names, so switching back is the same write path as switching to a router. It answers `409 EBUILTIN` to any update, delete, or attempt to shadow its name. Other errors: `400 EBADLABEL` / `EBADURL` (an endpoint must be `http:` or `https:`) / `EBADMODEL`, `404 ENOTFOUND`.
 
 **`POST /api/lanes/:id/router/apply`** — Body: `{ profile: "<name>" }` — is where a profile reaches disk. Only the name comes from the request; what is written, and where, is derived from the lane row: