### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

Authorization headers being sent to API gateway are received on lambdas as "empty string"

### Expected Behavior

_No response_

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

Run a API Gateway with lambda integration

### Environment

```markdown
- OS:
- LocalStack:
```


### Anything else?

_No response_
if modified_request_to_backend:
            if modified_request_to_backend.url:
                request_url = get_proxy_backend_url(
                    modified_request_to_backend.url, original_url=request_url
                )
            data_to_send = modified_request_to_backend.data
            if modified_request_to_backend.method:
                method_to_send = modified_request_to_backend.method
        
        headers_to_send.pop("Transfer-Encoding", None)
        response = requests.request(
            method_to_send,
            url=request_url,
            data=data_to_send,
            headers=headers_to_send,
            stream=True,
            verify=False,
        )
    
    
    not_consumed = not getattr(response, "_content_consumed", True)
    pass_raw = not_consumed or response.headers.get("content-encoding") in ["gzip"]
    if pass_raw and getattr(response, "raw", None):
        new_content = response.raw.read()
        if new_content:
            response._content = new_content
    
    for listener in listeners_outbound:
        updated_response = listener.return_response(
            method=method_to_send or handler_chain_request.method,