### 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_
)
    elif integration_type in ["HTTP_PROXY", "HTTP"]:
        if ":servicediscovery:" in uri:
            
            client = aws_stack.connect_to_service("servicediscovery")
            service_id = uri.split("/")[-1]
            instances = client.list_instances(ServiceId=service_id)["Instances"]
            instance = (instances or [None])[0]
            if instance and instance.get("Id"):
                uri = "http://%s/%s" % (instance["Id"], invocation_path.lstrip("/"))
        
        invocation_context.context = helpers.get_event_request_context(invocation_context)
        invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
        request_templates = RequestTemplates()
        payload = request_templates.render(invocation_context)
        if isinstance(payload, dict):
            payload = json.dumps(payload)
        uri = apply_request_parameters(
            uri,
            integration=integration,
            path_params=path_params,
            query_params=query_string_params,
        )
        result = requests.request(method=method, url=uri, data=payload, headers=headers)
        
        invocation_context.response = result
        response_templates = ResponseTemplates()
        response_templates.render(invocation_context)
        return invocation_context.response
    elif integration_type == "MOCK":