### 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_
API_ENDPOINT = os.environ.get("API_ENDPOINT") or "https://api.localstack.cloud/v1"
ANALYTICS_API = os.environ.get("ANALYTICS_API") or "https://analytics.localstack.cloud/v1"
LOCALSTACK_WEB_PROCESS = "LOCALSTACK_WEB_PROCESS"
LOCALSTACK_INFRA_PROCESS = "LOCALSTACK_INFRA_PROCESS"
AWS_REGION_US_EAST_1 = "us-east-1"
DEFAULT_LAMBDA_CONTAINER_REGISTRY = "lambci/lambda"
try:
    MAX_POOL_CONNECTIONS = int(os.environ["MAX_POOL_CONNECTIONS"])
except Exception:
    MAX_POOL_CONNECTIONS = 150
TEST_AWS_ACCESS_KEY_ID = "test"
TEST_AWS_SECRET_ACCESS_KEY = "test"
INTERNAL_AWS_ACCESS_KEY_ID = "__internal_call__"
INTERNAL_AWS_SECRET_ACCESS_KEY = "__internal_call__"
HEADER_LOCALSTACK_ACCOUNT_ID = "x-localstack-account-id"
LS_LOG_TRACE = "trace"
LS_LOG_TRACE_INTERNAL = "trace-internal"
TRACE_LOG_LEVELS = [LS_LOG_TRACE, LS_LOG_TRACE_INTERNAL]
OFFICIAL_IMAGES = [
    "localstack/localstack",
    "localstack/localstack-light",
    "localstack/localstack-full",
]
S3_VIRTUAL_HOSTNAME = "s3.%s" % LOCALHOST_HOSTNAME
S3_STATIC_WEBSITE_HOSTNAME = "s3-website.%s" % LOCALHOST_HOSTNAME
DEFAULT_DEVELOP_PORT = 5678
DEFAULT_BUCKET_MARKER_LOCAL = "__local__"
OS_USER_OPENSEARCH = "localstack"
READY_MARKER_OUTPUT = "Ready."