### 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_
def forward_request(
        self, method: str, path: str, data: MessagePayload, headers: Headers
    ) -> Optional[RoutingRequest]:
        """Return a RoutingRequest with modified request data, or None to forward the request
        unmodified"""
        return None
    def return_response(