feat(js-client-sdk): add ability to customize storage impl

[joker23]

This PR will enable browser sdk to take in a custom storage implementation.

Additional details:

• added common implementation for client sdks to eventually to make this feature common to all js client sdks
• custom storage implementations will be wrapped to ensure they don't throw

---

Note

Low Risk
Optional API with safe wrapper; default persistence path is unchanged when storage is not set.

Overview
Adds an optional storage setting on the browser (and React client) SDK so apps can plug in their own persistence instead of window.localStorage, while keeping today’s default when the option is omitted.

Shared client code introduces the LDStorage contract (get / set / clear) and createSafeStorage, which adapts custom implementations to the internal storage API: failures are logged, reads return null, and writes are dropped so a bad implementation cannot take down the host app. The browser client validates storage, wraps it with createSafeStorage, and passes it into BrowserPlatform, which uses the override when provided and otherwise still chooses LocalStorage only when supported.

^{Reviewed by Cursor Bugbot for commit a1f4117. Bugbot is set up for automated code reviews on this repo. Configure here.}

[github-actions]

@launchdarkly/js-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 26389 bytes
Compressed size limit: 29000
Uncompressed size: 129320 bytes

[github-actions]

@launchdarkly/js-client-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 38768 bytes
Compressed size limit: 39000
Uncompressed size: 212567 bytes

[github-actions]

@launchdarkly/browser size report
This is the brotli compressed size of the ESM build.
Compressed size: 179554 bytes
Compressed size limit: 200000
Uncompressed size: 831422 bytes

[github-actions]

@launchdarkly/js-client-sdk size report
This is the brotli compressed size of the ESM build.
Compressed size: 32002 bytes
Compressed size limit: 34000
Uncompressed size: 114243 bytes

[joker23]

@cursor review

[cursor]

✅ Bugbot reviewed your changes and found no new issues!

Comment @cursor review or bugbot run to trigger another review on this PR

^{Reviewed by Cursor Bugbot for commit 0b27576. Configure here.}

[devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no potential bugs to report.

View in Devin Review to see 3 additional findings.

[kinyoklion]

Should extend the 'logs no warnings when all configuration is valid' test with storage like the RN equivalent test.