### 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_
)
        
        path = path.replace("{test_param1}", "foo1")
        url = gateway_request_url(api_id=result["id"], stage_name=self.TEST_STAGE_NAME, path=path)
        data = {}
        for method in ("GET", "POST", "PUT", "PATCH", "DELETE", "OPTIONS"):
            body = json.dumps(data) if method in ("POST", "PUT", "PATCH") else None
            result = getattr(requests, method.lower())(url, data=body)
            if method != "DELETE":
                self.assertEqual(200, result.status_code)
                parsed_body = json.loads(to_str(result.content))
                self.assertEqual(method, parsed_body.get("httpMethod"))
            else:
                self.assertEqual(204, result.status_code)
    def test_apigateway_with_custom_authorization_method(self):
        apigw_client = aws_stack.create_external_boto_client("apigateway")
        # creat
        lambda_name = "apigw-lambda-%s" % short_uid()
        self.create_lambda_function(lambda_name)
        lambda_uri = aws_stack.lambda_function_arn(lambda_name)
        # creat
        api = apigw_client.create_rest_api(name="test-api", description="")
        api_id = api["id"]
        root_res_id = apigw_client.get_resources(restApiId=api_id)["items"][0]["id"]
        # creat
        authorizer = apigw_client.create_authorizer(
            restApiId=api_id,
            name="lambda_authorizer",
            type="TOKEN",
            authorizerUri="arn:aws:apigateway:us-east-1:lambda:path/ \