### 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_
mock_integration = MockIntegration()
        return mock_integration.invoke(invocation_context)
    if method == "OPTIONS":
        
        return get_cors_response(headers)
    raise Exception(
        'API Gateway integration type "%s", method "%s", URI "%s" not yet implemented'
        % (integration_type, method, uri)
    )
def apply_request_response_templates(
    data: Union[Response, bytes],
    templates: Dict[str, str],
    content_type: str = None,
    as_json: bool = False,
):
    """Apply the matching request/response template (if it exists) to the payload data and return the result"""
    content_type = content_type or APPLICATION_JSON
    is_response = isinstance(data, Response)
    templates = templates or {}
    template = templates.get(content_type)
    if not template:
        return data
    content = (data.content if is_response else data) or ""
    result = VtlTemplate().render_vtl(template, content, as_json=as_json)
    if is_response:
        data._content = result
        update_content_length(data)
        return data
    return result