feat(github): Pass GitHub App type to check run updates

[sentry]

fix(github-app): include appType in check run updates

Summary

Fixes KILOCODE-WEB-197X. The issue was that updateCheckRun omitted the appType parameter, causing it to default to incorrect GitHub App credentials and resulting in 502 errors from the GitHub API.

This change modifies check run update calls to explicitly include the GitHub App type. This ensures the correct GitHub App context is used when updating check runs, defaulting to 'standard' if not specified.

Verification

• Confirmed that check run updates no longer result in 502 errors from the GitHub API.
• Verified that check runs are updated successfully with the correct GitHub App context.

Visual Changes

N/A

Reviewer Notes

This fix was generated by Seer in Sentry, triggered by jean@kilocode.ai. 👁️ Run ID: 11481996

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[kilo-code-bot]

Code Review Summary

Status: No Issues Found | Recommendation: Merge

Files Reviewed (2 files)

• src/app/api/internal/code-review-status/[reviewId]/route.ts
• src/routers/code-reviews/code-reviews-router.ts

[jeanduplessis]

Due to the monorepo restructure you will need to recreate this PR on a new branch from main. Pass the prompt found at, https://github.com/Kilo-Org/cloud/blob/main/plans/monorepo-migration-prompt.md, to your coding agent while running in this branch. Please close this PR once done or if you don't plan to proceed with it.

[alex-alecu]

Continued #3087