fix(amazonq): update Q profile and customizations on language-servers crash restart

[floralph]

Problem

Investigation revealed that when the language server crashed and automatically restarted, the crash recovery mechanism only sent bearer tokens but failed to send profile configuration, leaving the language server stuck in PENDING_Q_PROFILE state.

Solution

Fixed the incomplete crash recovery mechanism by:
• Enhanced crash recovery handler to send both bearer token AND profile configuration during language server restart
• Added comprehensive logging with context tags ([startup], [crash-recovery]) for future debugging
• Created centralized authentication function to prevent code duplication between startup and crash recovery paths
• Added debug logging to pushConfigUpdate and sendProfileToLsp functions to track configuration values

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[github-actions]

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[yueny2020]

why we need this?

[floralph]

The GitHub checks for duplicates fails, but the code is similar, but not duplicates of what it finds.