### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

localstack v0.12.17 released with this enhancement that downcases all headers:

> canonicalize HTTP header names in API GW Lambda events to lower-case
> https://github.com/localstack/localstack/commit/657d9fe4bd0e3c4284e6fa67aec62d265d5d9fff

This does not match what we observe in production - custom headers are not downcased. See this part of the docs on API Gateway and how it canonicalizes headers:

> API Gateway enacts the following restrictions and limitations when handling methods with either Lambda integration or HTTP integration.
> * Header names and query parameters are processed in a case-sensitive way.
> * The following table lists the headers that may be dropped, remapped, or otherwise modified when sent to your integration endpoint or sent back by your integration endpoint. In this table: 
> https://docs.aws.amazon.com/apigateway/latest/developerguide/api-gateway-known-issues.html

### Expected Behavior

I would expect that localstack follows the same canonicalization rules that API Gateway implements.

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    docker run localstack/localstack:0.12.17

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Used [Insomnia](https://insomnia.rest/) client to issue a POST request to an AWS Lambda function via API Gateway with custom headers. For example, `X-A-Custom-Header`. Attempt to read the custom header in your lambda function case sensitively.


### Environment

```markdown
- OS: macOS for POST request + whatever the docker container uses for localstack
- LocalStack: 0.12.17
```


### Anything else?

_No response_
)
        # STEP 
        _prepare_method_integration(
            integr_kwargs={
                "requestTemplates": {
                    "application/json": """
                    #set($data = $util.escapeJavaScript($input.json('$')))
                    {"input": $data, "stateMachineArn": "%s"}
                    """
                    % sm_arn
                }
            }
        )
        # invok
        client.create_deployment(restApiId=rest_api["id"], stageName="dev")
        url = gateway_request_url(api_id=rest_api["id"], stage_name="dev", path="/")
        test_data = {"test": "test-value"}
        resp = requests.post(url, data=json.dumps(test_data))
        self.assertEqual(200, resp.status_code)
        self.assertIn("executionArn", resp.content.decode())
        self.assertIn("startDate", resp.content.decode())
        # STEP 
        _prepare_method_integration(overwrite=True)
        test_data_1 = {
            "input": json.dumps(test_data),
            "name": "MyExecution",
            "stateMachineArn": sm_arn,
        }
        # invok
        resp = requests.post(url, data=json.dumps(test_data_1))