### 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_
if "TEST_AWS_ACCOUNT_ID" not in os.environ:
    os.environ["TEST_AWS_ACCOUNT_ID"] = DEFAULT_AWS_ACCOUNT_ID
TEST_AWS_ACCOUNT_ID = os.environ["TEST_AWS_ACCOUNT_ID"]
TEST_AWS_REGION_NAME = "us-west-2"
MODULE_MAIN_PATH = os.path.dirname(os.path.realpath(__file__))
LOCALSTACK_ROOT_FOLDER = os.path.realpath(os.path.join(MODULE_MAIN_PATH, ".."))
LOCALSTACK_VENV_FOLDER = os.environ.get("VIRTUAL_ENV")
if not LOCALSTACK_VENV_FOLDER:
    
    LOCALSTACK_VENV_FOLDER = os.path.join(LOCALSTACK_ROOT_FOLDER, ".venv")
    if not os.path.isdir(LOCALSTACK_VENV_FOLDER):
        
        LOCALSTACK_VENV_FOLDER = os.path.realpath(
            os.path.join(LOCALSTACK_ROOT_FOLDER, "..", "..", "..")
        )
DEFAULT_VOLUME_DIR = "/var/lib/localstack"
PATH_USER_REQUEST = "_user_request_"
DOCKER_IMAGE_NAME = "localstack/localstack"
DOCKER_IMAGE_NAME_FULL = "localstack/localstack-full"
CONFIG_UPDATE_PATH = "/?_config_"
INTERNAL_RESOURCE_PATH = "/_localstack"
ENV_INTERNAL_TEST_RUN = "LOCALSTACK_INTERNAL_TEST_RUN"
ENV_INTERNAL_TEST_COLLECT_METRIC = "LOCALSTACK_INTERNAL_TEST_COLLECT_METRIC"
ENV_PRO_ACTIVATED = "PRO_ACTIVATED"
HEADER_CONTENT_TYPE = "Content-Type"
TEXT_XML = "text/xml"
APPLICATION_AMZ_JSON_1_0 = "application/x-amz-json-1.0"
APPLICATION_AMZ_JSON_1_1 = "application/x-amz-json-1.1"
APPLICATION_AMZ_CBOR_1_1 = "application/x-amz-cbor-1.1"
APPLICATION_CBOR = "application/cbor"