### 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_
if not stage:
        return {}
    region_name = [name for name, region in apigateway_backends.items() if api_id in region.apis][0]
    api_gateway_client = aws_stack.connect_to_service("apigateway", region_name=region_name)
    try:
        response = api_gateway_client.get_stage(restApiId=api_id, stageName=stage)
        return response.get("variables")
    except Exception:
        LOG.info(f"Failed to get stage {stage} for api id {api_id}")
        return {}
def get_event_request_context(invocation_context: ApiInvocationContext):
    method = invocation_context.method
    path = invocation_context.path
    headers = invocation_context.headers
    integration_uri = invocation_context.integration_uri
    resource_path = invocation_context.resource_path
    resource_id = invocation_context.resource_id
    set_api_id_stage_invocation_path(invocation_context)
    relative_path, query_string_params = extract_query_string_params(
        path=invocation_context.path_with_query_string
    )
    api_id = invocation_context.api_id
    stage = invocation_context.stage
    source_ip = headers.get("X-Forwarded-For", ",").split(",")[-2].strip()
    integration_uri = integration_uri or ""
    account_id = integration_uri.split(":lambda:path")[-1].split(":function:")[0].split(":")[-1]
    account_id = account_id or TEST_AWS_ACCOUNT_ID
    domain_name = f"{api_id}.execute-api.{LOCALHOST_HOSTNAME}"