### 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_
"""Adjust potential content lengths and checksums after modifying the response."""
        if response.headers and response.content:
            if "Content-Length" in response.headers:
                response.headers["Content-Length"] = str(len(to_bytes(response.content)))
            if "x-amz-crc32" in response.headers:
                response.headers["x-amz-crc32"] = calculate_crc32(response.content)
T = TypeVar("T", bound="RegionBackend")
class RegionBackend:
    """Base class for provider backends. RegionBackend lookup methods are not thread safe.
    RegionBackend transparently performs namespacing per AWS account IDs.
    Usage
    =====
    RegionBackend must be subclassed in the provider where it is to be used.
    Cross-region data must be defined as class attributes, and region-specific data as instance attributes.
        class FooBackend(RegionBackend):