### 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`
return validator["validateRequestBody"]
    @staticmethod
    def should_validate_request(validator):
        return validator.get("validateRequestParameters")
def run_authorizer(invocation_context: ApiInvocationContext, authorizer: Dict):
    
    pass
def authorize_invocation(invocation_context: ApiInvocationContext):
    client = aws_stack.connect_to_service("apigateway")
    authorizers = client.get_authorizers(restApiId=invocation_context.api_id, limit=100).get(
        "items", []
    )
    for authorizer in authorizers:
        run_authorizer(invocation_context, authorizer)
def validate_api_key(api_key: str, stage: str):
    usage_plan_ids = []
    client = aws_stack.connect_to_service("apigateway")
    usage_plans = client.get_usage_plans()
    for item in usage_plans.get("items", []):
        api_stages = item.get("apiStages", [])
        usage_plan_ids.extend(
            item.get("id") for api_stage in api_stages if api_stage.get("stage") == stage
        )
    for usage_plan_id in usage_plan_ids:
        usage_plan_keys = client.get_usage_plan_keys(usagePlanId=usage_plan_id)
        for key in usage_plan_keys.get("items", []):
            if key.get("value") == api_key:
                return True
    return False
def is_api_key_valid(is_api_key_required: bool, headers: Dict[str, str], stage: str):