### 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_
cloudformation = connect_to_service("cloudformation", region_name=region_name)
    stacks = cloudformation.describe_stacks(StackName=stack_name)
    for stack in stacks["Stacks"]:
        if stack["StackName"] == stack_name:
            return stack
def deploy_cf_stack(stack_name, template_body):
    cfn = connect_to_service("cloudformation")
    cfn.create_stack(StackName=stack_name, TemplateBody=template_body)
    
    return await_stack_completion(stack_name)
def await_stack_status(stack_name, expected_statuses, retries=20, sleep=2, region_name=None):
    def check_stack():
        stack = get_stack_details(stack_name, region_name=region_name)
        if stack["StackStatus"] not in expected_statuses:
            raise Exception(
                'Status "%s" for stack "%s" not in expected list: %s'
                % (stack["StackStatus"], stack_name, expected_statuses)
            )
        return stack
    expected_statuses = (
        expected_statuses if isinstance(expected_statuses, list) else [expected_statuses]
    )
    return retry(check_stack, retries, sleep)
def await_stack_completion(stack_name, retries=20, sleep=2, statuses=None, region_name=None):
    statuses = statuses or ["CREATE_COMPLETE", "UPDATE_COMPLETE", "DELETE_COMPLETE"]
    return await_stack_status(
        stack_name, statuses, retries=retries, sleep=sleep, region_name=region_name
    )