### 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`
dynamodb_client.delete_table(TableName=ddb_table)
            result = lambda_client.list_event_source_mappings(EventSourceArn=latest_stream_arn)
            assert 1 == len(result["EventSourceMappings"])
        finally:
            lambda_client.delete_event_source_mapping(UUID=event_source_mapping_uuid)
    def test_dynamodb_event_source_mapping_with_on_failure_destination_config(
        self,
        lambda_client,
        create_lambda_function,
        sqs_client,
        sqs_queue_arn,
        sqs_create_queue,
        create_iam_role_with_policy,
        dynamodb_client,
        dynamodb_create_table,
    ):
        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"
        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(