### 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_
relative_path, query_string_params = extract_query_string_params(path=invocation_path)
    integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
    integration_type = integration_type_orig.upper()
    uri = integration.get("uri") or integration.get("integrationUri") or ""
    
    
    custom_auth_header = invocation_context.headers.pop(HEADER_LOCALSTACK_AUTHORIZATION, "")
    invocation_context.headers["Authorization"] = custom_auth_header
    try:
        path_params = extract_path_params(path=relative_path, extracted_path=resource_path)
        invocation_context.path_params = path_params
    except Exception:
        path_params = {}
    if (uri.startswith("arn:aws:apigateway:") and ":lambda:path" in uri) or uri.startswith(
        "arn:aws:lambda"
    ):
        if integration_type == "AWS_PROXY":
            return LambdaProxyIntegration().invoke(invocation_context)
        elif integration_type == "AWS":
            return LambdaIntegration().invoke(invocation_context)
        raise Exception(
            f'API Gateway integration type "{integration_type}", action "{uri}", method "{method}"'
        )
    elif integration_type == "AWS":
        if "kinesis:action/" in uri:
            if uri.endswith("kinesis:action/PutRecord"):
                target = kinesis_listener.ACTION_PUT_RECORD
            elif uri.endswith("kinesis:action/PutRecords"):
                target = kinesis_listener.ACTION_PUT_RECORDS