### 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_
return make_error_response(
            "Unable to find integration for: %s %s (%s)" % (method, invocation_path, raw_path),
            404,
        )
    
    invocation_context.resource_path = extracted_path
    invocation_context.integration = method_integration
    return invoke_rest_api_integration(invocation_context)
def invoke_rest_api_integration(invocation_context: ApiInvocationContext):
    try:
        response = invoke_rest_api_integration_backend(invocation_context)
        
        
        invocation_context.response = response
        response = apply_response_parameters(invocation_context)
        return response
    except Exception as e:
        msg = f"Error invoking integration for API Gateway ID '{invocation_context.api_id}': {e}"
        LOG.exception(msg)
        return make_error_response(msg, 400)
def invoke_rest_api_integration_backend(invocation_context: ApiInvocationContext):
    
    invocation_path = invocation_context.path_with_query_string
    method = invocation_context.method
    data = invocation_context.data
    headers = invocation_context.headers
    resource_path = invocation_context.resource_path
    integration = invocation_context.integration
    integration_response = integration.get("integrationResponses", {})
    response_templates = integration_response.get("200", {}).get("responseTemplates", {})