### 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_
name,
    description=None,
    resources=None,
    stage_name=None,
    enabled_api_keys=None,
    env=None,
    usage_plan_name=None,
    region_name=None,
    auth_creator_func=None,  
):
    if enabled_api_keys is None:
        enabled_api_keys = []
    client = connect_to_service("apigateway", env=env, region_name=region_name)
    resources = resources or []
    stage_name = stage_name or "testing"
    usage_plan_name = usage_plan_name or "Basic Usage"
    description = description or 'Test description for API "%s"' % name
    LOG.info('Creating API resources under API Gateway "%s".', name)
    api = client.create_rest_api(name=name, description=description)
    api_id = api["id"]
    auth_id = None
    if auth_creator_func:
        auth_id = auth_creator_func(api_id)
    resources_list = client.get_resources(restApiId=api_id)
    root_res_id = resources_list["items"][0]["id"]
    
    for path, methods in resources.items():
        
        parent_id = root_res_id
        for path_part in path.split("/"):