test(nuxt): cover the client plugin's pages-router instrumentation

[61465]

Summary

Closes #13813.

Fills in the Nuxt client-plugin unit tests requested by @mydea. Picks up where the dormant draft #24583 (@vansh-nagar, inactive since 2026-02; pinged by @JPeer264 on 2026-10-12 with no reply within the stated window) left off and extends it to all five code paths in packages/nuxt/src/runtime/plugins/sentry.client.ts.

What this covers

1. Vue route provider setup (_INTERNAL_createVueRouteProvider → setRouteProvider)
   • Installs when getClient() returns a client, $router is on the Nuxt app, and no provider is set
   • Respects a user-supplied provider (does not override)
   • Skips without an initialized client
   • Skips without a $router on the Nuxt app
2. browserTracingIntegration
   • Adds with { router, routeLabel: 'path' } on the happy path
   • Skips without an initialized client
   • Skips without a $router
3. app:created hook — Vue integration
   • Calls vueIntegration with attachErrorHandler: false and the live Vue app
   • Handles the client-torn-down case (second getClient() returns undefined)
4. app:error hook — status-code filtering
   • Nuxt error + 4xx/3xx status → skip
   • Nuxt error + 5xx → report
   • Non-Nuxt error → report regardless of status shape
5. vue:error hook
   • Forwards { error, instance, info } to reportNuxtError

Pattern

Mocks mirror packages/nuxt/test/runtime/hooks/captureErrorHook.test.ts:

• vi.hoisted(() => vi.fn()) for cross-module mock refs
• vi.mock(import('…'), async importOriginal => …) for @sentry/core, @sentry/vue, nuxt/app
• vi.spyOn(utils, 'reportNuxtError') for the local ../utils module

Pure vitest, no native dependencies. Runs under the existing yarn test command in packages/nuxt/.

Local verification

Syntax parse via tsc --noEmit --skipLibCheck on the test file passes (only "Cannot find module" warnings remain for workspace packages, which CI resolves via yarn install). I was unable to materialise the Yarn/npm workspace node_modules locally on my Windows + Node 24 shell (ETARGET on whatwg-url@^17.2.0, same error reported upstream in jsdom's current chain), so I'm relying on your CI matrix to confirm the suite is green across the vitest + jsdom environment configured in packages/nuxt/vite.config.ts.

Notes

• All five code paths were taken straight from the current src/runtime/plugins/sentry.client.ts; happy to re-shape the test file structure if the preferred house style here is one describe per public behaviour rather than per source-code region.
• @vansh-nagar's draft is credited above; if they prefer to carry this over I'll close this PR and move my coverage there as review comments.

cc @mydea + @JPeer264 — thanks for the pre-blessing on #13813 and for keeping the issue open.

🤖 Used Claude Code for the test scaffolding and to cross-check the mock shapes against existing nuxt tests; every assertion was signed off by me on content.