[Nuxt] Follow-up tasks

[s1gr1d]

Important

This is the tracking issue for any tasks we still want to do in the Nuxt SDK. Even tough the SDK is officially still in beta, it's safe to use it in production.

General open issues, related to the Nuxt SDK are collected as standalone issues with Package: nuxt

General Sentry Nuxt SDK Tracking Issue: #9095

Overall, the following things should work:

• running the dev mode without problems (but server-side monitoring can be limited in dev)
• building + running the project instruments client + server

### Tasklist
- [x] Nuxt 4 Support https://github.com/getsentry/sentry-javascript/pull/13799
- [ ] https://github.com/getsentry/sentry-javascript/issues/14029
- [x] CLI Wizard https://github.com/getsentry/sentry-wizard/pull/719
- [ ] pnpm fix: https://github.com/getsentry/sentry-javascript/issues/14221
- [ ] https://github.com/getsentry/sentry-javascript/issues/13330
- [ ] Nuxt 4: Put server-side config in server and client-side config in app