### 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_
invocation_path = invocation_context.path_with_query_string
    raw_path = invocation_context.path or invocation_path
    method = invocation_context.method
    headers = invocation_context.headers
    
    authorize_invocation(invocation_context)
    extracted_path, resource = helpers.get_target_resource_details(invocation_context)
    if not resource:
        return make_error_response("Unable to find path %s" % invocation_context.path, 404)
    
    validator = RequestValidator(invocation_context, aws_stack.connect_to_service("apigateway"))
    if not validator.is_request_valid():
        return make_error_response("Invalid request body", 400)
    api_key_required = resource.get("resourceMethods", {}).get(method, {}).get("apiKeyRequired")
    if not is_api_key_valid(api_key_required, headers, invocation_context.stage):
        return make_error_response("Access denied - invalid API key", 403)
    resource_methods = resource.get("resourceMethods", {})
    resource_method = resource_methods.get(method, {})
    if not resource_method:
        
        
        resource_method = resource_methods.get("ANY", {}) or resource_methods.get(
            "X-AMAZON-APIGATEWAY-ANY-METHOD", {}
        )
    method_integration = resource_method.get("methodIntegration")
    if not method_integration:
        if method == "OPTIONS" and "Origin" in headers:
            
            return get_cors_response(headers)