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).

client/src/components/lanes/__tests__/LaneCard.test.tsx

@@ -27,10 +27,14 @@ vi.mock("../../../lib/api", () => ({
       agentsInstall: vi.fn(),
       mcpSync: vi.fn(),
       syncBaseCheck: vi.fn(),
+      routerApply: vi.fn(),
     },
     locks: {
       list: vi.fn(),
     },
+    aiRouters: {
+      list: vi.fn(),
+    },
   },
 }));
 
@@ -43,6 +47,10 @@ beforeEach(() => {
   vi.mocked(api.lanes.runtime).mockResolvedValue({ available: false });
   vi.mocked(api.lanes.integration).mockReset();
   vi.mocked(api.lanes.integration).mockResolvedValue({ enabled: false });
+  // Only the built-in profile by default: a user who never opened the AI Router
+  // tab has nothing to switch between, so the row stays out of every other test.
+  vi.mocked(api.aiRouters.list).mockReset();
+  vi.mocked(api.aiRouters.list).mockResolvedValue({ items: [] });
 });
 vi.mocked(api.locks.list).mockReset();
 vi.mocked(api.locks.list).mockResolvedValue({ locks: [] });
@@ -440,3 +448,63 @@ describe("LaneToolsMenu", () => {
     expect(container).toBeEmptyDOMElement();
   });
 });
+
+describe("LaneCard — the AI Router picker", () => {
+  const native = {
+    name: "native",
+    label: "Native (Claude / Codex)",
+    builtin: true,
+    baseUrl: "",
+    hasKey: false,
+    model: "claude-sonnet-5",
+    models: {
+      opus: "claude-opus-5",
+      sonnet: "claude-sonnet-5",
+      haiku: "claude-haiku-4-5-20251001",
+      smallFast: "claude-haiku-4-5-20251001",
+      subagent: "claude-haiku-4-5-20251001",
+    },
+  };
+  const nineRouter = { ...native, name: "9router", label: "9Router", builtin: false };
+
+  it("stays out of the card when native is the only profile", async () => {
+    vi.mocked(api.aiRouters.list).mockResolvedValue({ items: [native] });
+    render(<LaneCard lane={makeLane()} onAction={vi.fn()} />);
+    await waitFor(() => expect(api.aiRouters.list).toHaveBeenCalled());
+    expect(screen.queryByTestId("lane-router-1")).toBeNull();
+  });
+
+  it("shows the lane's current profile selected", async () => {
+    vi.mocked(api.aiRouters.list).mockResolvedValue({ items: [native, nineRouter] });
+    render(<LaneCard lane={makeLane({ router_profile: "9router" })} onAction={vi.fn()} />);
+    expect(await screen.findByDisplayValue("9Router")).toBeInTheDocument();
+  });
+
+  it("applies a switch through the router route, not a lane PATCH", async () => {
+    vi.mocked(api.aiRouters.list).mockResolvedValue({ items: [native, nineRouter] });
+    vi.mocked(api.lanes.routerApply).mockResolvedValue({
+      tool: "claude",
+      file: "/work/demo/.claude/settings.local.json",
+      profile: "9router",
+    });
+    render(<LaneCard lane={makeLane()} onAction={vi.fn()} />);
+
+    await userEvent.selectOptions(await screen.findByDisplayValue("Native (Claude / Codex)"), [
+      "9router",
+    ]);
+    await waitFor(() => expect(api.lanes.routerApply).toHaveBeenCalledWith(1, "9router"));
+    expect(await screen.findByDisplayValue("9Router")).toBeInTheDocument();
+  });
+
+  it("puts the selection back when the apply fails", async () => {
+    vi.mocked(api.aiRouters.list).mockResolvedValue({ items: [native, nineRouter] });
+    vi.mocked(api.lanes.routerApply).mockRejectedValue(new Error("EBADJSON"));
+    render(<LaneCard lane={makeLane()} onAction={vi.fn()} />);
+
+    await userEvent.selectOptions(await screen.findByDisplayValue("Native (Claude / Codex)"), [
+      "9router",
+    ]);
+    await waitFor(() => expect(screen.getByText("EBADJSON")).toBeInTheDocument());
+    expect(screen.getByDisplayValue("Native (Claude / Codex)")).toBeInTheDocument();
+  });
+});