Ensure Sentry is able to initialize without DSN if Spotlight is enabled

### Description

As part of the [requirement to make Spotlight compatible with SDKs](https://github.com/getsentry/spotlight/issues/545), the SDKs should be able to initialize without a DSN if Spotlight is enabled

```[tasklist]
### 🛑 Blocked by
- [ ] https://github.com/getsentry/sentry-cocoa/issues/4620
- [ ] https://github.com/getsentry/sentry-java/issues/3981
```

JS already supports this https://github.com/getsentry/sentry-javascript/blob/938fe65a1bb363bf92cb35df24edea27d3d75883/packages/core/src/baseclient.ts#L322

```[tasklist]
### Tasks
- [ ] Check if JS layer can send events to Spotlight without changes on native
```

Start with packages/core/src/baseclient.ts in the linked Sentry JavaScript implementation and compare its no-DSN behavior with the React Native JS layer. Check whether events can reach Spotlight without native changes, while noting the linked Cocoa and Java issues as blockers. Done means Spotlight-enabled initialization works without a DSN.

Written by the indexing model from the issue text.