### 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_
from werkzeug.exceptions import HTTPException
from localstack import config
from localstack.aws.accounts import get_aws_account_id
from localstack.config import (
    EXTRA_CORS_ALLOWED_HEADERS,
    EXTRA_CORS_ALLOWED_ORIGINS,
    EXTRA_CORS_EXPOSE_HEADERS,
    MULTI_ACCOUNTS,
)
from localstack.constants import (
    APPLICATION_JSON,
    AWS_REGION_US_EAST_1,
    BIND_HOST,
    HEADER_LOCALSTACK_AUTHORIZATION,
    HEADER_LOCALSTACK_REQUEST_URL,
)
from localstack.http.request import get_full_raw_path
from localstack.services.messages import Headers, MessagePayload
from localstack.services.messages import Request as RoutingRequest
from localstack.services.messages import Response as RoutingResponse
from localstack.utils.asyncio import run_sync
from localstack.utils.aws import aws_stack
from localstack.utils.aws.aws_responses import LambdaResponse, calculate_crc32
from localstack.utils.aws.aws_stack import is_internal_call_context
from localstack.utils.aws.request_context import RequestContextManager, get_proxy_request_for_thread
from localstack.utils.crypto import generate_ssl_cert
from localstack.utils.functions import empty_context_manager
from localstack.utils.json import json_safe
from localstack.utils.net import wait_for_port_open
from localstack.utils.server import http2_server