### 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_
return to_response_json("account", data, self_link="/account")
def to_vpc_link_response_json(data):
    return to_response_json("vpclink", data)
def to_client_cert_response_json(data):
    return to_response_json("clientcertificate", data, id_attr="clientCertificateId")
def to_response_json(model_type, data, api_id=None, self_link=None, id_attr=None):
    if isinstance(data, list) and len(data) == 1:
        data = data[0]
    id_attr = id_attr or "id"
    result = common.clone(data)
    if not self_link:
        self_link = "/%ss/%s" % (model_type, data[id_attr])
        if api_id:
            self_link = "/restapis/%s/%s" % (api_id, self_link)
    if "_links" not in result:
        result["_links"] = {}
    result["_links"]["self"] = {"href": self_link}
    result["_links"]["curies"] = {
        "href": "https://docs.aws.amazon.com/apigateway/latest/developerguide/restapi-authorizer-latest.html",
        "name": model_type,
        "templated": True,
    }
    result["_links"]["%s:delete" % model_type] = {"href": self_link}
    return result
def find_api_subentity_by_id(api_id, entity_id, map_name):
    region_details = APIGatewayRegion.get()
    auth_list = getattr(region_details, map_name).get(api_id) or []
    entity = ([a for a in auth_list if a["id"] == entity_id] or [None])[0]
    return entity
def gateway_request_url(api_id, stage_name, path):