### 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_
response = invocation_context.response
    integration = invocation_context.integration
    int_responses = integration.get("integrationResponses") or {}
    if not int_responses:
        return response
    entries = list(int_responses.keys())
    return_code = str(response.status_code)
    if return_code not in entries:
        if len(entries) > 1:
            LOG.info("Found multiple integration response status codes: %s", entries)
            return response
        return_code = entries[0]
    response_params = int_responses[return_code].get("responseParameters", {})
    for key, value in response_params.items():
        
        if str(key).lower().startswith("method.response.header."):
            header_name = key[len("method.response.header.") :]
            response.headers[header_name] = value.strip("'")
    return response
def invoke_rest_api_from_request(invocation_context: ApiInvocationContext):
    helpers.set_api_id_stage_invocation_path(invocation_context)
    try:
        return invoke_rest_api(invocation_context)
    except AuthorizationError as e:
        api_id = invocation_context.api_id
        return make_error_response("Not authorized to invoke REST API %s: %s" % (api_id, e), 403)
def invoke_rest_api(invocation_context: ApiInvocationContext):