feat: AI Router per session; Settings at the sidebar's foot; shipped tool icons

AI Router is a session's choice, not a workspace's:
- an AI Router select beside the model picker in the workspace home and a
  pane's run form (RouterPicker.tsx; hidden off Claude/Codex or with only
  Native; last pick remembered in agenthub.session.router)
- `router` on POST /api/lanes/:id/start and POST /api/run; Claude gets
  --settings=<data dir>/router-settings/<name>.json (0600, outranks project
  and user settings, key never in argv), Codex -p <name> + key via tmux -e
- dashboard_runs.router; a resume naming none goes back to it
- profile PATCH rewrites the session files, DELETE removes them
- gone: POST /api/lanes/:id/router/apply, LaneRouterRow, the Add-workspace
  router radio, the claude-/codex-home `routers` answers; clearLaneRouters
  takes the old keys out of each workspace's settings.local.json at start

Sidebar: Settings pinned to the foot, the collapse toggle beside it (stacked
on the rail) instead of in the brand's corner.

Icons: shipped SVG marks for claude, codex, openclaw and hermes (a
black/white pair swapped by theme) in client/public/ai-tools/, used by
AiToolIcon before the server's favicon.

server/routes/run.js

@@ -30,6 +30,7 @@ const tmux = require("../lib/tmux");
 const lanesLib = require("../lib/lanes");
 const { sameOriginGuard } = require("../lib/same-origin");
 const aiTools = require("../lib/ai-tools");
+const { resumeRouter } = require("../lib/router-apply");
 
 const router = Router();
 
@@ -321,8 +322,12 @@ router.post("/", (req, res) => {
       laneId,
       cwd,
       tool: lane?.tool || "claude",
-      // The lane's own router, exactly like its tool — never a body field.
-      routerProfile: lane?.router_profile || null,
+      // The session's AI Router, by name (key and endpoint from the store); a
+      // resume that names none goes back to the one its session last ran on.
+      router:
+        body.router !== undefined
+          ? body.router
+          : resumeRouter(typeof body.resumeSessionId === "string" ? body.resumeSessionId : null),
       model: typeof body.model === "string" && body.model ? body.model : null,
       permissionMode:
         typeof body.permissionMode === "string" && ALLOWED_PERMISSION_MODES.has(body.permissionMode)