refactor(logging): preserve explicit LiteLLM logger levels

[fcogidi]

Summary

Preserves explicit LiteLLM caller-configured logging level when constructing LMClient, while keeping infermesh quiet by default.

Clickup Ticket(s): N/A

Type of Change

• 🐛 Bug fix (non-breaking change that fixes an issue)
• ✨ New feature (non-breaking change that adds functionality)
• 💥 Breaking change
• 📝 Documentation update
• 🔧 Refactoring
• ⚡ Performance improvement
• 🧪 Test improvements
• 🔒 Security fix

Changes Made

• Preserved caller-configured LiteLLM verbose logger levels, including DEBUG.
• Kept the default quiet behavior by setting LiteLLM loggers to WARNING only when unconfigured.
• Added regression tests for debug preservation and default logger behavior.

Testing

• Tests pass locally (uv run pytest tests/)
• Type checking passes (uv run mypy <src_dir>)
• Linting passes (uv run ruff check src_dir/)
• Manual testing performed (describe below)

Manual testing details:

• Ran uv run pytest tests/test_client_runtime.py successfully.
• Verified litellm._turn_on_debug() before LMClient(...) now emits LiteLLM DEBUG traces during a failing local request.

Screenshots/Recordings

N/A

Related Issues

N/A

Deployment Notes

No deployment steps or migrations required.

Checklist

• Code follows the project's style guidelines
• Self-review of code completed
• Documentation updated (if applicable)
• No sensitive information (API keys, credentials) exposed