### 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_
"""Check whether a valid region is provided, and if not then raise an Exception."""
    auth_header = headers.get("Authorization")
    if not auth_header:
        raise Exception('Unable to find "Authorization" header in request')
    replaced = re.sub(r".*Credential=([^,]+),.*", r"\1", auth_header)
    if auth_header == replaced:
        raise Exception('Unable to find "Credential" section in "Authorization" header')
    
    
    parts = replaced.split("/")
    region = parts[2]
    if region not in config.VALID_REGIONS:
        raise Exception('Invalid region specified in "Authorization" header: "%s"' % region)
def set_default_region_in_headers(headers, service=None, region=None):
    
    
    if not config.USE_SINGLE_REGION:
        return
    auth_header = headers.get("Authorization")
    region = region or get_region()
    if not auth_header:
        if service:
            headers["Authorization"] = mock_aws_request_headers(service, region_name=region)[
                "Authorization"
            ]
        return
    replaced = re.sub(r"(.*Credential=[^/]+/[^/]+/)([^/])+/", r"\1%s/" % region, auth_header)
    headers["Authorization"] = replaced
def fix_account_id_in_arns(response, colon_delimiter=":", existing=None, replace=None):