### 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_
APPLICATION_JSON = "application/json"
APPLICATION_XML = "application/xml"
APPLICATION_OCTET_STREAM = "application/octet-stream"
APPLICATION_X_WWW_FORM_URLENCODED = "application/x-www-form-urlencoded"
HEADER_ACCEPT_ENCODING = "Accept-Encoding"
TRUE_STRINGS = ("1", "true", "True")
FALSE_STRINGS = ("0", "false", "False")
LOG_LEVELS = ("trace-internal", "trace", "debug", "info", "warn", "error", "warning")
ELASTICSEARCH_DEFAULT_VERSION = "Elasticsearch_7.10"
ELASTICSEARCH_PLUGIN_LIST = [
    "analysis-icu",
    "ingest-attachment",
    "analysis-kuromoji",
    "mapper-murmur3",
    "mapper-size",
    "analysis-phonetic",
    "analysis-smartcn",
    "analysis-stempel",
    "analysis-ukrainian",
]
ELASTICSEARCH_DELETE_MODULES = ["ingest-geoip"]
OPENSEARCH_DEFAULT_VERSION = "OpenSearch_1.1"
OPENSEARCH_PLUGIN_LIST = [
    "ingest-attachment",
]
ELASTICMQ_JAR_URL = (
    "https://s3-eu-west-1.amazonaws.com/softwaremill-public/elasticmq-server-1.1.0.jar"
)
STEPFUNCTIONS_ZIP_URL = "https://s3.amazonaws.com/stepfunctionslocal/StepFunctionsLocal.zip"
KMS_URL_PATTERN = "https://s3-eu-west-2.amazonaws.com/local-kms/3/local-kms_<arch>.bin"