### 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`
TEST_LAMBDA_PYTHON,
    TEST_LAMBDA_PYTHON_ECHO,
    TEST_LAMBDA_PYTHON_UNHANDLED_ERROR,
)
TEST_STAGE_NAME = "testing"
TEST_SNS_TOPIC_NAME = "sns-topic-1"
THIS_FOLDER = os.path.dirname(os.path.realpath(__file__))
TEST_LAMBDA_PARALLEL_FILE = os.path.join(THIS_FOLDER, "functions", "lambda_parallel.py")
lambda_role = {
    "Version": "2012-10-17",
    "Statement": [
        {
            "Effect": "Allow",
            "Principal": {"Service": "lambda.amazonaws.com"},
            "Action": "sts:AssumeRole",
        }
    ],
}
s3_lambda_permission = {
    "Version": "2012-10-17",
    "Statement": [
        {
            "Effect": "Allow",
            "Action": [
                "sqs:*",
                "dynamodb:DescribeStream",
                "dynamodb:GetRecords",
                "dynamodb:GetShardIterator",
                "dynamodb:ListStreams",
                "kinesis:DescribeStream",