fix(cliproxy): use management auth-url route in paste callback

tests/unit/cliproxy/auth-types-management-path.test.ts

@@ -0,0 +1,19 @@
+import { describe, expect, it } from 'bun:test';
+import { getManagementAuthUrlPath } from '../../../src/cliproxy/auth/auth-types';
+
+describe('auth-types management auth-url path', () => {
+  it('maps providers to CLIProxyAPI management auth-url routes', () => {
+    expect(getManagementAuthUrlPath('gemini')).toBe(
+      '/v0/management/gemini-cli-auth-url?is_webui=true'
+    );
+    expect(getManagementAuthUrlPath('codex')).toBe('/v0/management/codex-auth-url?is_webui=true');
+    expect(getManagementAuthUrlPath('agy')).toBe(
+      '/v0/management/antigravity-auth-url?is_webui=true'
+    );
+    expect(getManagementAuthUrlPath('claude')).toBe(
+      '/v0/management/anthropic-auth-url?is_webui=true'
+    );
+    expect(getManagementAuthUrlPath('ghcp')).toBe('/v0/management/github-auth-url?is_webui=true');
+    expect(getManagementAuthUrlPath('kiro')).toBe('/v0/management/kiro-auth-url?is_webui=true');
+  });
+});