### 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_
test_port = get_free_tcp_port()
        backend_url = "http://localhost:%s%s" % (test_port, self.API_PATH_HTTP_BACKEND)
        
        proxy = self.start_http_backend(test_port)
        
        result = self.connect_api_gateway_to_http(
            int_type, "test_gateway2", backend_url, path=self.API_PATH_HTTP_BACKEND
        )
        url = gateway_request_url(
            api_id=result["id"],
            stage_name=self.TEST_STAGE_NAME,
            path=self.API_PATH_HTTP_BACKEND,
        )
        
        origin = "localhost"
        result = requests.options(url, headers={"origin": origin})
        self.assertEqual(result.status_code, 200)
        self.assertTrue(
            re.match(result.headers["Access-Control-Allow-Origin"].replace("*", ".*"), origin)
        )
        self.assertIn("POST", result.headers["Access-Control-Allow-Methods"])
        self.assertIn("PATCH", result.headers["Access-Control-Allow-Methods"])
        custom_result = json.dumps({"foo": "bar"})
        
        result = requests.get(url)
        self.assertEqual(200, result.status_code)
        expected = custom_result if int_type == "custom" else "{}"
        self.assertEqual(expected, json.loads(to_str(result.content))["data"])