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.

client/src/components/lanes/WorkspaceHome.tsx

@@ -39,6 +39,7 @@ import { fmtCost, timeAgo } from "../../lib/format";
 import { useAiTools } from "../../hooks/useAiTools";
 import { useProjectTreePrefs } from "../../lib/projectTreePrefs";
 import { ModelPicker, PromptEditor } from "../run/RunSetup";
+import { RouterPicker, routerArg, useSessionRouter } from "../run/RouterPicker";
 import { Select } from "../Select";
 import { MarkdownPreview } from "../MarkdownPreview";
 import { WorkspaceRowMenu, type WorkspaceMenuAction } from "./WorkspaceRowMenu";
@@ -107,6 +108,7 @@ function HomeBody({
   // Claude Desktop's worktree toggle: the new session gets a git worktree of
   // its own (`claude --worktree`), made and cleaned up by Claude Code.
   const [inWorktree, setInWorktree] = useState(false);
+  const [router, setRouter] = useSessionRouter();
   const [starting, setStarting] = useState(false);
   const [error, setError] = useState<string | null>(null);
   const [liveRuns, setLiveRuns] = useState<RunHandle[]>([]);
@@ -231,6 +233,7 @@ function HomeBody({
         ...(model ? { model } : {}),
         ...(claude ? { permissionMode: mode } : {}),
         ...(claude && inWorktree && project?.root ? { worktree: true } : {}),
+        ...routerArg(aiTools.byId(lane.tool)?.routerCapable, router),
       });
       setPrompt("");
       const runId = r.run_id ?? paneRunId(lane.id, slot);
@@ -341,6 +344,7 @@ function HomeBody({
             <div className="w-56">
               <ModelPicker value={model} onChange={setModel} tool={lane.tool} />
             </div>
+            <RouterPicker tool={lane.tool} value={router} onChange={setRouter} />
             {claude && (
               <div className="w-44">
                 <Select<PermissionMode>