feat: read from custom-llm-provider header

[tlecomte]

Title

Read the custom llm provider option from the request header "custom-llm-provider".

Relevant issues

Fixes #15522

Pre-Submission checklist

Please complete all items before asking a LiteLLM maintainer to review your PR

• I have Added testing in the tests/litellm/ directory, Adding at least 1 test is a hard requirement - see details
• I have added a screenshot of my new test passing locally
• My PR passes all unit tests on make test-unit
• My PR's scope is as isolated as possible, it only solves 1 specific problem

Type

🆕 New Feature

Changes

Several LiteLLM operations require passing a custom_llm_provider entry in the request. Some requests need this parameter in the request body, others in a query string. Here we add the ability to pass this parameter in a http header instead.

Passing this parameter in a header is easier compared to passing it in the request body or as a query parameter:

1. It would be consistent across all request types
2. The Python and the .NET client for OpenAI both allow to add headers easily
3. Headers are easy to manipulate in a proxy

[vercel]

@tlecomte is attempting to deploy a commit to the CLERKIEAI Team on Vercel.

A member of the Team first needs to authorize it.

[krrishdholakia]

can you document this flow? @tlecomte