Fix MCP response write ordering - #2652

Summary

• Flush MCP stdio responses explicitly before the server proceeds to subsequent input.
• Defer response-related MCP diagnostics until after the response write attempt, including parse, oversized, auth, protocol, rate-limit, and tool-error paths.
• Add regression coverage for response/log ordering and write-failure handling.

Validation

• dotnet test tests/CodeIndex.Tests/CodeIndex.Tests.csproj -c Release -p:UseSharedCompilation=false --filter "FullyQualifiedName~McpServerTests"
• dotnet run --project tools/CodeIndex.Changelog -- check
• git diff --check
• adversarial review: No blocking/actionable issues found.

Documentation / Changelog

• Added changelog.d/unreleased/1927.fixed.md.
• No README/guide changes were needed; this is an MCP transport/error-ordering fix covered by tests and changelog.

Follow-up candidates

• Local .cdidx index recovery: cdidx reports Last batch did not complete; rerun dotnet ./src/CodeIndex/bin/Debug/net8.0/cdidx.dll index --rebuild or a full rebuild when convenient.

Fixes #1927
Fixes #2020
Fixes #2021