### 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_
f"{aws_stack.render_velocity_template(template, data)}&QueueUrl={queue_url}"
                    )
                headers = aws_stack.mock_aws_request_headers(service="sqs", region_name=region_name)
                url = urljoin(config.service_url("sqs"), f"{TEST_AWS_ACCOUNT_ID}/{queue}")
                result = common.make_http_request(
                    url, method="POST", headers=headers, data=new_request
                )
                return result
        raise Exception(
            'API Gateway AWS integration action URI "%s", method "%s" not yet implemented'
            % (uri, method)
        )
    elif integration_type == "AWS_PROXY":
        if uri.startswith("arn:aws:apigateway:") and ":dynamodb:action" in uri:
            
            table_name = uri.split(":dynamodb:action")[1].split("&Table=")[1]
            action = uri.split(":dynamodb:action")[1].split("&Table=")[0]
            if "PutItem" in action and method == "PUT":
                response_template = response_templates.get("application/json")
                if response_template is None:
                    msg = "Invalid response template defined in integration response."
                    LOG.info("%s Existing: %s", msg, response_templates)
                    return make_error_response(msg, 404)
                response_template = json.loads(response_template)
                if response_template["TableName"] != table_name:
                    msg = "Invalid table name specified in integration response template."
                    return make_error_response(msg, 404)
                dynamo_client = aws_stack.connect_to_resource("dynamodb")
                table = dynamo_client.Table(table_name)
                event_data = {}