### 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_
bucket, object_key = uri_match.group("bucket", "object")
                LOG.debug("Getting request for bucket %s object %s", bucket, object_key)
                try:
                    object = s3.get_object(Bucket=bucket, Key=object_key)
                except s3.exceptions.NoSuchKey:
                    msg = "Object %s not found" % object_key
                    LOG.debug(msg)
                    return make_error_response(msg, 404)
                headers = aws_stack.mock_aws_request_headers(service="s3")
                if object.get("ContentType"):
                    headers["Content-Type"] = object["ContentType"]
                
                response = request_response_stream(stream=object["Body"], headers=headers)
                return response
            else:
                msg = "Request URI does not match s3 specifications"
                LOG.warning(msg)
                return make_error_response(msg, 400)
        if method == "POST":
            if uri.startswith("arn:aws:apigateway:") and ":sqs:path" in uri:
                template = integration["requestTemplates"][APPLICATION_JSON]
                account_id, queue = uri.split("/")[-2:]
                region_name = uri.split(":")[3]
                if "GetQueueUrl" in template or "CreateQueue" in template:
                    new_request = (
                        f"{aws_stack.render_velocity_template(template, data)}&QueueName={queue}"
                    )
                else:
                    queue_url = f"{config.get_edge_url()}/{account_id}/{queue}"
                    new_request = (