### 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`
@classmethod
    def check_passthrough_behavior(cls, passthrough_behavior: str, request_template: str):
        return MappingTemplates(passthrough_behavior).check_passthrough_behavior(request_template)
    def invoke(self, invocation_context: ApiInvocationContext) -> Response:
        passthrough_behavior = invocation_context.integration.get("passthroughBehavior") or ""
        request_template = invocation_context.integration.get("requestTemplates", {}).get(
            invocation_context.headers.get(HEADER_CONTENT_TYPE)
        )
        
        
        try:
            self.check_passthrough_behavior(passthrough_behavior, request_template)
        except MappingTemplates.UnsupportedMediaType:
            http_status = HTTPStatus(415)
            return MockIntegration._create_response(
                http_status.value,
                headers={"Content-Type": APPLICATION_JSON},
                data=json.dumps({"message": f"{http_status.phrase}"}),
            )
        
        request_payload = self.request_templates.render(invocation_context)