fix(sliding-sync): refresh stale room member display names when global profile differs

[eleboucher]

Description

Fixes #1520

Type of change

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• This change requires a documentation update

Checklist:

• My code follows the style guidelines of this project
• I have performed a self-review of my own code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• My changes generate no new warnings

AI disclosure:

• Partially AI assisted (clarify which code was AI assisted and briefly explain what it does).
• Fully AI generated (explain what all the generated code does in moderate detail).

[github-actions]

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Status	Preview URL	Commit	Alias	Updated (UTC)
✅ Deployment successful!	https://pr-1530-sable.raspy-dream-bb1d.workers.dev	eee1198	pr-1530	Fri, 31 Jul 2026 20:29:37 GMT