### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

Beacuse we use localstack for alot of integration tests and we often need to mock some rest api I wanted to use the mock function of api gateway to make a simple rest api mock, but I cant get it to work properly. 

When calling my api i get a empty dictionary response:
```
curl http://localhost:4566/restapis/kiyhgeyswi/test/_user_request_/123-v
* Uses proxy env variable no_proxy == 'localhost'
*   Trying 127.0.0.1:4566...
* Connected to localhost (127.0.0.1) port 4566 (#0)
> GET /restapis/kiyhgeyswi/test/_user_request_/test HTTP/1.1
> Host: localhost:4566
> User-Agent: curl/7.79.1
> Accept: */*
>
* Mark bundle as not supporting multiuse
< HTTP/1.1 200
< Content-Type: application/json
< Content-Length: 2
< Access-Control-Allow-Origin: *
< Access-Control-Allow-Methods: HEAD,GET,PUT,POST,DELETE,OPTIONS,PATCH
< Access-Control-Allow-Headers: authorization,cache-control,content-length,content-md5,content-type,etag,location,x-amz-acl,x-amz-content-sha256,x-amz-date,x-amz-request-id,x-amz-security-token,x-amz-tagging,x-amz-target,x-amz-user-agent,x-amz-version-id,x-amzn-requestid,x-localstack-target,amz-sdk-invocation-id,amz-sdk-request
< Access-Control-Expose-Headers: etag,x-amz-version-id
< Connection: close
< date: Tue, 22 Feb 2022 08:29:28 GMT
< server: hypercorn-h11
<
{}
```

### Expected Behavior

It should respond with the mock response. When deploying the exact same api in aws a get this response:
```
curl https://$API_ID.execute-api.$REGION.amazonaws.com/dev/123
{
    "statusCode": 200,
    "id2": 123
}
```

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker-compose up
```
  version: '3.3'
  services:
    localstack:
      image: localstack/localstack:latest
      container_name: localstack_tests5
      ports:
        - '4566:4566'
        - '4571:4571'
        - '8080:8080'
      environment:
        - DOCKER_HOST=unix:///var/run/docker.sock
        - SERVICES=kms,dynamodb,apigateway
        - DEBUG=1
        - DEFAULT_REGION=eu-west-1
        - DATA_DIR=/tmp/localstack/data
        - LAMBDA_EXECUTOR=local
        - KINESIS_PROVIDER=kinesalite
        - KMS_PROVIDER=local-kms
      volumes:
        - '/var/run/docker.sock:/var/run/docker.sock'
 ```

./create_mock_api.sh

 ```
#!/bin/bash
NAME=main
FILE=main.zip
HANDLER=main
REGION=us-east-1

#Create rest api
aws apigateway create-rest-api --region $REGION --name 'mockapi' --endpoint-url=http://localhost:4566

API_ID=$(aws apigateway get-rest-apis --query "items[?name==\`mockapi\`].id" --output text --region $REGION --endpoint-url=http://localhost:4566)
PARENT_RESOURCE_ID=$(aws apigateway get-resources --rest-api-id ${API_ID} --query 'items[0].id' --output text --region $REGION --endpoint-url=http://localhost:4566)

#Create resource associated with path
aws apigateway create-resource \
--region $REGION \
--rest-api-id $API_ID \
--parent-id $PARENT_RESOURCE_ID \
--path-part "{id}" \
--endpoint-url=http://localhost:4566

RESOURCEID=$(aws apigateway get-resources --rest-api-id $API_ID --region $REGION --endpoint-url=http://localhost:4566 --query 'items[?path==`/{id}`].id' --output text)

aws apigateway create-model \
--region $REGION \
--rest-api-id $API_ID \
--name Empty2 \
--content-type application/json \
--schema "{ \"$schema\": \"http://json-schema.org/draft-04/schema#\", \"title\": \"Empty Schema2\", \"type\": \"object\" }" \
--endpoint-url=http://localhost:4566

#Create get method and method response
aws apigateway put-method \
--region $REGION \
--rest-api-id $API_ID \
--resource-id $RESOURCEID \
--http-method GET \
--authorization-type "NONE" \
--request-parameters "method.request.path.id=true" \
--endpoint-url=http://localhost:4566

aws apigateway put-method-response \
--region $REGION \
--rest-api-id $API_ID \
--resource-id $RESOURCEID \
--http-method GET \
--status-code 200 \
--response-models "application/json=Empty2" \
--endpoint-url=http://localhost:4566

#Integrate api
aws apigateway put-integration \
--region $REGION \
--rest-api-id $API_ID \
--resource-id $RESOURCEID \
--http-method GET \
--type MOCK \
--passthrough-behavior WHEN_NO_MATCH \
--request-templates '{ "application/json": "{\"statusCode\": 200}" }' \
--endpoint-url=http://localhost:4566

aws apigateway put-integration-response \
--region $REGION \
--rest-api-id $API_ID \
--resource-id $RESOURCEID \
--http-method GET \
--status-code 200 \
--response-templates '{"application/json": "{\"statusCode\": 200,\"id2\": $input.params().path.id}"}' \
--endpoint-url=http://localhost:4566

#deploy
aws apigateway create-deployment \
 --region $REGION \
 --rest-api-id $API_ID \
 --stage-name test \
 --endpoint-url=http://localhost:4566
 
curl http://localhost:4566/restapis/${API_ID}/test/_user_request_/test -v
 ```

### Environment

```markdown
- OS: Windows 10
- LocalStack:  0.14.0 (latest/9cfaf4f1c73a)
```


### Anything else?

_No response_
import base64
import datetime
import json
import logging
import re
import time
from typing import Any, Dict, Optional, Tuple, Union
from urllib.parse import urljoin
import pytz
import requests
from flask import Response as FlaskResponse
from jsonschema import ValidationError, validate
from moto.apigateway.models import apigateway_backends
from requests.models import Response
from localstack import config
from localstack.constants import (
    APPLICATION_JSON,
    HEADER_LOCALSTACK_EDGE_URL,
    LOCALHOST_HOSTNAME,