### 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_
import json
import logging
import re
from typing import Any, Dict, List, Optional, Tuple
from urllib import parse as urlparse
from botocore.utils import InvalidArnException
from jsonpatch import apply_patch
from jsonpointer import JsonPointerException
from moto.apigateway import models as apigateway_models
from moto.apigateway.utils import create_id as create_resource_id
from requests.models import Response
from localstack import config
from localstack.constants import APPLICATION_JSON, PATH_USER_REQUEST, TEST_AWS_ACCOUNT_ID
from localstack.services.generic_proxy import RegionBackend
from localstack.utils import common
from localstack.utils.aws import aws_stack
from localstack.utils.aws.aws_responses import requests_error_response_json, requests_response
from localstack.utils.aws.aws_stack import parse_arn
LOG = logging.getLogger(__name__)
PATH_REGEX_MAIN = r"^/restapis/([A-Za-z0-9_\-]+)/[a-z]+(\?.*)?"
PATH_REGEX_SUB = r"^/restapis/([A-Za-z0-9_\-]+)/[a-z]+/([A-Za-z0-9_\-]+)/.*"
PATH_REGEX_SUB = r"^/restapis/([A-Za-z0-9_\-]+)/[a-z]+/([A-Za-z0-9_\-]+)/.*"
PATH_REGEX_AUTHORIZERS = r"^/restapis/([A-Za-z0-9_\-]+)/authorizers/?([^?/]+)?(\?.*)?"
PATH_REGEX_VALIDATORS = r"^/restapis/([A-Za-z0-9_\-]+)/requestvalidators/?([^?/]+)?(\?.*)?"