### 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_
region_name = region_name or get_region()
    pattern = re.sub(r"\([^\|]+\|.+\)", type, pattern)
    result = pattern.replace(".*", "%s") % (region_name, account_id, entity_name)
    if version:
        result = "%s:%s" % (result, version)
    return result
def lambda_function_name(name_or_arn):
    if ":" in name_or_arn:
        arn = parse_arn(name_or_arn)
        if arn["service"] != "lambda":
            raise ValueError("arn is not a lambda arn %s" % name_or_arn)
        return parse_arn(name_or_arn)["resource"].split(":")[1]
    else:
        return name_or_arn
def state_machine_arn(name, account_id=None, region_name=None):
    pattern = "arn:aws:states:%s:%s:stateMachine:%s"
    return _resource_arn(name, pattern, account_id=account_id, region_name=region_name)
def stepfunctions_activity_arn(name, account_id=None, region_name=None):
    pattern = "arn:aws:states:%s:%s:activity:%s"
    return _resource_arn(name, pattern, account_id=account_id, region_name=region_name)
def fix_arn(arn):
    """Function that attempts to "canonicalize" the given ARN. This includes converting
    resource names to ARNs, replacing incorrect regions, account IDs, etc."""
    if arn.startswith("arn:aws:lambda"):
        parts = arn.split(":")
        region = parts[3] if parts[3] in config.VALID_REGIONS else get_region()
        return lambda_function_arn(lambda_function_name(arn), region_name=region)
    LOG.warning("Unable to fix/canonicalize ARN: %s", arn)
    return arn
def cognito_user_pool_arn(user_pool_id, account_id=None, region_name=None):