### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

I am creating an API Gateway with a Lambda Authorizer via a CloudFormation stack. The stack specifies a data mapping from the request body passed to the API to the request body expected by Kinesis (search `RequestTemplates` in the attached CloudFormation template). It also specifies a model and a validator which are used to validate the request body. Both of these aspects of the API work in AWS itself but appear to be ignored in LocalStack. 

If I test with the same request body I passed to AWS, LocalStack throws an error indicating that the PutRecord call to Kinesis isn't providing a PartitionKey field. The data mapping should've taken care of it. If I then invoke the API in LocalStack with what *should* be sent to Kinesis, the invocation succeeds. This second invocation *should* violate the validation logic in the template, but doesn't. 

### Expected Behavior

The data mapping should correctly map the passed-in request body to a request body which Kinesis can understand. In addition, the validator should reject requests that don't pass validation.

### How are you starting LocalStack?

With a `docker run` command

### Steps To Reproduce

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

docker run --rm -it -p 4566:4566 -p 4571:4571 -e LOCALSTACK_API_KEY=$env:LOCALSTACK_API_KEY localstack/localstack

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

```
# create s3 bucket and upload c# lambda authorizer zip to s3
aws --endpoint-url=http://localhost:4566 s3api create-bucket --bucket timeseries-ingest-lambda-artifacts
aws --endpoint-url=http://localhost:4566 s3 cp index.zip s3://timeseries-ingest-lambda-artifacts/KinesisAPILambdaAuthorizer/index.zip
# create stream
aws --endpoint-url=http://localhost:4566 kinesis create-stream --stream-name test-timeseries-ingest-stream --shard-count 1
# create stack
aws --endpoint-url=http://localhost:4566 cloudformation create-stack --stack-name test-kinesis-api --template-body file://cloudformation-kinesis-api-resources-PER-ENVIRONMENT.yaml --parameters ParameterKey=AppName,ParameterValue=timeseries-ingest ParameterKey=Environment,ParameterValue=test
```

This is the CloudFormation template: [cloudformation-kinesis-api-resources-PER-ENVIRONMENT.yaml.txt](https://github.com/localstack/localstack/files/7973188/cloudformation-kinesis-api-resources-PER-ENVIRONMENT.yaml.txt)

### Environment

```markdown
- OS: Ubuntu 20.04
- LocalStack: latest (as of 3 days ago)
```


### Anything else?

_No response_
if not is_api_key_required:
        return True
    api_key = headers.get("X-API-Key")
    if not api_key:
        return False
    return validate_api_key(api_key, stage)
def update_content_length(response: Response):
    if response and response.content is not None:
        response.headers["Content-Length"] = str(len(response.content))
def apply_request_parameters(
    uri: str, integration: Dict[str, Any], path_params: Dict[str, str], query_params: Dict[str, str]
):
    request_parameters = integration.get("requestParameters")
    uri = uri or integration.get("uri") or integration.get("integrationUri") or ""
    if request_parameters:
        for key in path_params:
            
            request_param_key = f"integration.request.path.{key}"
            request_param_value = f"method.request.path.{key}"
            if request_parameters.get(request_param_key, None) == request_param_value:
                uri = uri.replace(f"{{{key}}}", path_params[key])
    if integration.get("type") != "HTTP_PROXY" and request_parameters:
        for key in query_params.copy():
            request_query_key = f"integration.request.querystring.{key}"
            request_param_val = f"method.request.querystring.{key}"
            if request_parameters.get(request_query_key, None) != request_param_val:
                query_params.pop(key)
    return add_query_params_to_url(uri, query_params)
def apply_template(