### 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_
# This 
            apigw_client.get_integration(
                restApiId=api_id, resourceId=api_resource["id"], httpMethod="GET"
            )
        self.assertEqual(ctx.exception.response["Error"]["Code"], "NotFoundException")
        
        lambda_client = aws_stack.create_external_boto_client("lambda")
        lambda_client.delete_function(FunctionName=lambda_name)
        apigw_client.delete_rest_api(restApiId=api_id)
    def test_api_gateway_handle_domain_name(self):
        domain_name = "%s.example.com" % short_uid()
        apigw_client = aws_stack.create_external_boto_client("apigateway")
        rs = apigw_client.create_domain_name(domainName=domain_name)
        self.assertEqual(200, rs["ResponseMetadata"]["HTTPStatusCode"])
        rs = apigw_client.get_domain_name(domainName=domain_name)
        self.assertEqual(200, rs["ResponseMetadata"]["HTTPStatusCode"])
        self.assertEqual(domain_name, rs["domainName"])
        
        apigw_client.delete_domain_name(domainName=domain_name)
    def _test_api_gateway_lambda_proxy_integration_any_method(self, fn_name, path):
        self.create_lambda_function(fn_name)
        
        lambda_uri = aws_stack.lambda_function_arn(fn_name)
        target_uri = aws_stack.apigateway_invocations_arn(lambda_uri)
        result = testutil.connect_api_gateway_to_http_with_lambda_proxy(
            "test_gateway3",
            target_uri,
            methods=["ANY"],
            path=path,
            stage_name=self.TEST_STAGE_NAME,