chore: migrate google-cloud-billing-budgets to librarian (#8547)

* chore: migrate google-cloud-billing-budgets to librarian
* add keep
* restore
* tidy

---------

Co-authored-by: Joe Wang <106995533+JoeWang1127@users.noreply.github.com>
Co-authored-by: JoeWang1127 <joewa@google.com>

Author: quirogas

librarian.yaml

@@ -437,7 +437,8 @@ libraries:
       - path: google/cloud/billing/budgets/v1
       - path: google/cloud/billing/budgets/v1beta1
     copyright_year: "2026"
-    skip_generate: true
+    keep:
+      - system-test/system.js
     nodejs:
       default_version: v1
   - name: google-cloud-binaryauthorization

packages/google-cloud-billing-budgets/.OwlBot.yaml

@@ -1,16 +1,16 @@
 {
-  "client_documentation": "https://cloud.google.com/nodejs/docs/reference/billing-budgets/latest",
-  "api_id": "billingbudgets.googleapis.com",
-  "distribution_name": "@google-cloud/billing-budgets",
-  "release_level": "stable",
-  "default_version": "v1",
-  "language": "nodejs",
-  "name_pretty": "Billing Budgets",
-  "repo": "googleapis/google-cloud-node",
-  "product_documentation": "https://cloud.google.com/billing/docs/how-to/budget-api-overview ",
-  "requires_billing": true,
-  "name": "billingbudgets",
-  "issue_tracker": "https://issuetracker.google.com/savedsearches/559770",
-  "api_shortname": "billingbudgets",
-  "library_type": "GAPIC_AUTO"
-}
+    "api_description": "The Cloud Billing Budget API stores Cloud Billing budgets, which define a\nbudget plan and the rules to execute as spend is tracked against that\nplan.",
+    "api_id": "billingbudgets.googleapis.com",
+    "api_shortname": "billingbudgets",
+    "client_documentation": "https://cloud.google.com/nodejs/docs/reference/billing-budgets/latest",
+    "default_version": "v1",
+    "distribution_name": "@google-cloud/billing-budgets",
+    "issue_tracker": "https://issuetracker.google.com/savedsearches/559770",
+    "language": "nodejs",
+    "library_type": "GAPIC_AUTO",
+    "name": "billingbudgets",
+    "name_pretty": "Cloud Billing Budget",
+    "product_documentation": "https://cloud.google.com/billing/docs",
+    "release_level": "stable",
+    "repo": "googleapis/google-cloud-node"
+}

packages/google-cloud-billing-budgets/.repo-metadata.json

@@ -29,4 +29,4 @@ describe('Integration Tests', () => {
       assert.include(err.message, 'PERMISSION_DENIED');
     }
   });
-});
+});