feat: show agent badge on workspace list (#23453)

- Adds `GET /api/experimental/chats/by-workspace` endpoint that returns
workspace_id → latest chat_id mapping
- Modifies FE to fetch this alongside the workspace list, gated on
`agents` experiment and render an "Agent" badge similar to the existing
"Task" badge in `WorkspacesTable`
- Badge links to the "latest chat" linked to the given workspace.

Notes:
- Intentionally uses `fetchWithPostFilter` for RBAC to decouple from
workspaces API — will migrate to `workspaces_expanded` view later.
- If users have multiple chats linked to the same workspace, the badge
will link to the most recently updated one.

> 🤖 This PR was created with the help of Coder Agents, and has been
reviewed by my human. 🧑‍💻

codersdk/chats.go

@@ -2028,6 +2028,26 @@ func (c *ExperimentalClient) GetMyChatUsageLimitStatus(ctx context.Context) (Cha
 	return resp, json.NewDecoder(res.Body).Decode(&resp)
 }
 
+// GetChatsByWorkspace returns a mapping of workspace ID to the latest
+// non-archived chat ID for each requested workspace. Workspaces with
+// no chats are omitted from the response.
+func (c *ExperimentalClient) GetChatsByWorkspace(ctx context.Context, workspaceIDs []uuid.UUID) (map[uuid.UUID]uuid.UUID, error) {
+	ids := make([]string, 0, len(workspaceIDs))
+	for _, id := range workspaceIDs {
+		ids = append(ids, id.String())
+	}
+	res, err := c.Request(ctx, http.MethodGet, fmt.Sprintf("/api/experimental/chats/by-workspace?workspace_ids=%s", strings.Join(ids, ",")), nil)
+	if err != nil {
+		return nil, err
+	}
+	defer res.Body.Close()
+	if res.StatusCode != http.StatusOK {
+		return nil, ReadBodyAsError(res)
+	}
+	var result map[uuid.UUID]uuid.UUID
+	return result, json.NewDecoder(res.Body).Decode(&result)
+}
+
 func formatChatStreamResponseError(response Response) string {
 	message := strings.TrimSpace(response.Message)
 	detail := strings.TrimSpace(response.Detail)