### 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`
):
        def check_logs():
            expected = [
                r'.*"Records":.*',
                r'.*"dynamodb": {(.*)}.*',
                r'.*"eventSource": ("aws:dynamodb").*',
                r'.*"eventName": ("INSERT").*',
                r'.*"Keys": {0}.*'.format(json.dumps(db_item)),
            ]
            check_lambda_logs(function_name, expected_lines=expected)
        function_name = f"lambda_func-{short_uid()}"
        role = f"test-lambda-role-{short_uid()}"
        policy_name = f"test-lambda-policy-{short_uid()}"
        table_name = f"test-table-{short_uid()}"
        partition_key = "my_partition_key"
        db_item = {partition_key: {"S": "hello world"}}
        try:
            role_arn = create_iam_role_with_policy(
                RoleName=role,
                PolicyName=policy_name,
                RoleDefinition=lambda_role,
                PolicyDefinition=s3_lambda_permission,
            )
            create_lambda_function(
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                func_name=function_name,
                runtime=LAMBDA_RUNTIME_PYTHON37,
                role=role_arn,
            )
            dynamodb_create_table(table_name=table_name, partition_key=partition_key)