### 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_
def test_extract_query_params(self):
        path, query_params = apigateway_listener.extract_query_string_params(
            "/foo/bar?foo=foo&bar=bar&bar=baz"
        )
        self.assertEqual("/foo/bar", path)
        self.assertEqual({"foo": "foo", "bar": ["bar", "baz"]}, query_params)
    def test_extract_path_params(self):
        params = apigateway_listener.extract_path_params("/foo/bar", "/foo/{param1}")
        self.assertEqual({"param1": "bar"}, params)
        params = apigateway_listener.extract_path_params("/foo/bar1/bar2", "/foo/{param1}/{param2}")
        self.assertEqual({"param1": "bar1", "param2": "bar2"}, params)
        params = apigateway_listener.extract_path_params("/foo/bar", "/foo/bar")
        self.assertEqual({}, params)
        params = apigateway_listener.extract_path_params("/foo/bar/baz", "/foo/{proxy+}")
        self.assertEqual({"proxy": "bar/baz"}, params)
    def test_path_matches(self):
        path, details = apigateway_listener.get_resource_for_path("/foo/bar", {"/foo/{param1}": {}})
        self.assertEqual("/foo/{param1}", path)
        path, details = apigateway_listener.get_resource_for_path(
            "/foo/bar", {"/foo/bar": {}, "/foo/{param1}": {}}
        )
        self.assertEqual("/foo/bar", path)
        path, details = apigateway_listener.get_resource_for_path(
            "/foo/bar/baz", {"/foo/bar": {}, "/foo/{proxy+}": {}}
        )
        self.assertEqual("/foo/{proxy+}", path)
        path, details = apigateway_listener.get_resource_for_path(
            "/foo/bar/baz", {"/{proxy+}": {}, "/foo/{proxy+}": {}}
        )
        self.assertEqual("/foo/{proxy+}", path)