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/run/RunSetup.tsx

@@ -59,6 +59,7 @@ import type {
 import type { Session, LaneTool } from "../../lib/types";
 import { useAiTools } from "../../hooks/useAiTools";
 import { Select } from "../Select";
+import { RouterPicker, routerArg, useSessionRouter } from "./RouterPicker";
 
 export interface SlashCommand {
   name: string;
@@ -409,6 +410,8 @@ export function RunSetup(props: RunSetupProps) {
   const canResume = !catalogTool || !!entry?.resume;
   // Per run, never remembered: the form starts unticked every time it mounts.
   const [bypass, setBypass] = useState(false);
+  const [router, setRouter] = useSessionRouter();
+  const routerSent = routerArg(entry?.routerCapable, router).router;
   const canBypass = props.tool !== "claude" && !!entry?.bypass;
   // Claude and Codex always take a model; a catalog tool only when its file names the flag.
   const hasModel = !catalogTool || !!entry?.modelFlag;
@@ -450,6 +453,15 @@ export function RunSetup(props: RunSetupProps) {
               <ModelPicker value={props.model} onChange={props.onModelChange} tool={props.tool} />
             </Field>
           )}
+          {/* A resume goes back to the router its session ran on (server-side). */}
+          {!props.resumeSession && (
+            <RouterPicker
+              tool={props.tool}
+              value={router}
+              onChange={setRouter}
+              wrap={(select) => <Field label={t("lanes:router.label")}>{select}</Field>}
+            />
+          )}
           {catalogTool ? null : props.tool === "codex" ? (
             <>
               <Field label={t("fields.sandbox")}>
@@ -536,7 +548,7 @@ export function RunSetup(props: RunSetupProps) {
                   // update has not landed yet on this tick, so reading
                   // props.resumeSession here would resume nothing.
                   if (s && !props.busy)
-                    handleStart(props, s, bypass, hasModel, entry?.defaultModel);
+                    handleStart(props, s, bypass, hasModel, entry?.defaultModel, routerSent);
                 }}
                 // Fall back to the typed cwd when no lane is selected — otherwise
                 // an unfiltered picker lists every session from every repo.
@@ -564,7 +576,14 @@ export function RunSetup(props: RunSetupProps) {
           value={props.prompt}
           onChange={props.onPromptChange}
           onSubmit={() =>
-            handleStart(props, undefined, canBypass && bypass, hasModel, entry?.defaultModel)
+            handleStart(
+              props,
+              undefined,
+              canBypass && bypass,
+              hasModel,
+              entry?.defaultModel,
+              routerSent
+            )
           }
           placeholder={t("fields.promptPlaceholderTerminal", { tool: aiTools.label(props.tool) })}
           rows={5}
@@ -620,7 +639,14 @@ export function RunSetup(props: RunSetupProps) {
         </div>
         <button
           onClick={() =>
-            handleStart(props, undefined, canBypass && bypass, hasModel, entry?.defaultModel)
+            handleStart(
+              props,
+              undefined,
+              canBypass && bypass,
+              hasModel,
+              entry?.defaultModel,
+              routerSent
+            )
           }
           disabled={
             !props.binaryFound ||
@@ -655,7 +681,8 @@ function handleStart(
   session?: Session,
   bypass?: boolean,
   hasModel?: boolean,
-  defaultModel?: string | null
+  defaultModel?: string | null,
+  router?: string
 ) {
   const resume = session ?? props.resumeSession;
   const codex = props.tool === "codex";
@@ -682,6 +709,8 @@ function handleStart(
     // asked for, so a resume carries no prompt at all.
     initialPrompt: resume ? undefined : props.prompt || undefined,
     bypass: bypass || undefined,
+    // A resume names none: the server goes back to its session's own router.
+    router: resume ? undefined : router,
   });
 }