### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

I Created an REST APIGatway with lambda handler using AWS CDK. when I hit the API the response header content type is always  set to "text/html"  instead of `application/json` . even if I changed the response template or added the header to lambda returned response. when I deploy the same stack on AWS account  the response header is fine and work as expected

### Expected Behavior

Response Header content-type to be `application/json`

### How are you starting LocalStack?

  `localstack start` command

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command`)

    `localstack start` command

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

I'm using AWS CDK typescript application and deploy it using `cdklocal` command on localstack

Using CDK follow the steps:
- create  REST ApiGateway
- create Lambda function 
- integrate lambda with APIGatway

On Command line:
- deploy using `cdklocal deploy` command

Finally
- test the endpoint and check the response header content-type




### Environment

```markdown
- OS: Windows 
- LocalStack: latest
```


### Anything else?

When I deploy the same stack on AWS account everything works fine and the response header content-type is `application/json`
import json
import logging
import re
from typing import Any, Dict, Union
from urllib.parse import urljoin
import requests
from flask import Response as FlaskResponse
from jsonschema import ValidationError, validate
from requests.models import Response
from localstack import config
from localstack.constants import (
    APPLICATION_JSON,
    HEADER_LOCALSTACK_AUTHORIZATION,
    TEST_AWS_ACCOUNT_ID,
)
from localstack.services.apigateway import helpers
from localstack.services.apigateway.context import ApiInvocationContext
from localstack.services.apigateway.helpers import (
    extract_path_params,
    extract_query_string_params,
    get_cors_response,
    make_error_response,
)
from localstack.services.apigateway.integration import (