### 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_
"""Return URL for inbound API gateway for given API ID, stage name, and path"""
    pattern = "%s/restapis/{api_id}/{stage_name}/%s{path}" % (
        config.service_url("apigateway"),
        PATH_USER_REQUEST,
    )
    return pattern.format(api_id=api_id, stage_name=stage_name, path=path)
def tokenize_path(path):
    return path.lstrip("/").split("/")
def extract_path_params(path: str, extracted_path: str) -> Dict[str, str]:
    tokenized_extracted_path = tokenize_path(extracted_path)
    
    path_params_list = [(i, v) for i, v in enumerate(tokenized_extracted_path) if "{" in v]
    tokenized_path = tokenize_path(path)
    path_params = {}
    for param in path_params_list:
        path_param_name = param[1][1:-1]
        path_param_position = param[0]
        if path_param_name.endswith("+"):
            path_params[path_param_name.rstrip("+")] = "/".join(
                tokenized_path[path_param_position:]
            )
        else:
            path_params[path_param_name] = tokenized_path[path_param_position]
    path_params = common.json_safe(path_params)
    return path_params
def extract_query_string_params(path: str) -> Tuple[str, Dict[str, str]]: