### 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_
try:
            listener_result = listener.forward_request(
                method=handler_chain_request.method,
                path=handler_chain_request.path,
                data=handler_chain_request.data,
                headers=handler_chain_request.headers,
            )
        except HTTPException as e:
            
            return http_exception_to_response(e)
        if isinstance(listener, MessageModifyingProxyListener):
            if isinstance(listener_result, RoutingRequest):
                
                handler_chain_request.method = (
                    listener_result.method or handler_chain_request.method
                )
                handler_chain_request.path = listener_result.path or handler_chain_request.path
                if listener_result.data is not None:
                    handler_chain_request.data = listener_result.data
                if listener_result.headers is not None:
                    handler_chain_request.headers = listener_result.headers
            continue
        if isinstance(listener_result, Response):
            response = listener_result
            break
        if isinstance(listener_result, LambdaResponse):
            response = listener_result
            break
        if isinstance(listener_result, dict):
            response = Response()