### 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`
"kinesis:DescribeStreamSummary",
                "kinesis:GetRecords",
                "kinesis:GetShardIterator",
                "kinesis:ListShards",
                "kinesis:ListStreams",
                "kinesis:SubscribeToShard",
                "logs:CreateLogGroup",
                "logs:CreateLogStream",
                "logs:PutLogEvents",
            ],
            "Resource": ["*"],
        }
    ],
}
class TestSQSEventSourceMapping:
    @pytest.mark.skip_snapshot_verify
    def test_event_source_mapping_default_batch_size(
        self,
        create_lambda_function,
        lambda_client,
        sqs_client,
        sqs_create_queue,
        sqs_queue_arn,
        dynamodb_client,
        dynamodb_create_table,
        lambda_su_role,
        snapshot,
    ):
        snapshot.add_transformer(snapshot.transform.lambda_api())
        function_name = f"lambda_func-{short_uid()}"