### 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`
WHEN_NO_MATCH = "WHEN_NO_MATCH"
    WHEN_NO_TEMPLATES = "WHEN_NO_TEMPLATES"
    NEVER = "NEVER"
class MappingTemplates:
    """
    API Gateway uses mapping templates to transform incoming requests before they are sent to the
    integration back end. With API Gateway, you can define one mapping template for each possible
    content type. The content type selection is based on the Content-Type header of the incoming
    request. If no content type is specified in the request, API Gateway uses an application/json
    mapping template. By default, mapping templates are configured to simply pass through the
    request input. Mapping templates use Apache Velocity to generate a request to your back end.
    """
    passthrough_behavior: PassthroughBehavior
    class UnsupportedMediaType(Exception):