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/LANES.md

@@ -43,7 +43,7 @@ The modal has a segmented toggle mirroring the CLI's two modes:
 - **Repo** — adopts the given directory as-is (maps to `agenthub lanes add --cwd`). No branch fields; the auto-setup summary above does not run (adopting is instant, nothing to wait on).
 - **Worktree** (default) — provisions a managed worktree (maps to `agenthub lanes add --repo`). Unlike the CLI, the modal requires you to type the new branch's name yourself rather than deriving one from the title — the underlying route still derives one when `branch` is omitted, so the CLI's behavior is unchanged.
 
-Under the *Pipeline template* select, a **Model endpoint** radio chooses between **Native** (the vendor's own API) and **AI Router** — a profile saved in Settings › AI Router (9Router, OpenRouter, a self-hosted gateway). Choosing one writes it into the new lane's own config as the lane is created: a claude lane gets the `env` block of its `.claude/settings.local.json`, a codex lane gets a `$CODEX_HOME/<name>.config.toml` layer plus a `-p <name>` on every spawn. The UI offers the choice only at creation, and with no profile saved the radio stays on Native and says so. Unlike the lane's tool, though, this one is not immutable: `POST /api/lanes/:id/router/apply` re-points an existing lane, rewriting its config and its `router_profile` column together. A session already running in the pane keeps the endpoint it started with — Claude Code reads `settings.local.json` at startup — so the change takes effect on the next run. Editing or deleting the profile itself in Settings also reaches every lane using it: an edit re-applies it (so a rotated key actually lands), a delete resets those lanes to Native. See `docs/API.md` § AI Router profiles.
+Under the *Pipeline template* select, a **Model endpoint** radio chooses between **Native** (the vendor's own API) and **AI Router** — a profile saved in Settings › AI Router (9Router, OpenRouter, a self-hosted gateway). Choosing one writes it into the new lane's own config as the lane is created: a claude lane gets the `env` block of its `.claude/settings.local.json`, a codex lane gets a `$CODEX_HOME/<name>.config.toml` layer plus a `-p <name>` on every spawn. With no profile saved the radio stays on Native and says so. Unlike the lane's tool, this one is not immutable: an existing lane's card in Workspace carries an **AI Router** select showing its current profile, and picking another calls `POST /api/lanes/:id/router/apply`, rewriting the lane's config and its `router_profile` column together. (The select is hidden when `native` is the only profile — there would be nothing to choose.) A failed apply puts the select back rather than leaving the card claiming a router the lane did not get. A session already running in the pane keeps the endpoint it started with — Claude Code reads `settings.local.json` at startup — so the change takes effect on the next run. Editing or deleting the profile itself in Settings also reaches every lane using it: an edit re-applies it (so a rotated key actually lands), a delete resets those lanes to Native. See `docs/API.md` § AI Router profiles.
 
 Under the *Pipeline template* select, a **Create/update readme file** checkbox (Claude lanes only — Codex has no slash commands) starts the new lane's terminal run right after creation with `/sg-all:sg-md-writing` as its first message, so the lane opens already writing/refreshing its `CLAUDE.md`/`AGENTS.md` set. It requires the `sg-all` plugin to be installed in Claude Code; without it the session simply reports an unknown command. Leave it unticked to create the lane without starting a run.