### 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`
def invoke(self, invocation_context: ApiInvocationContext) -> Response:
        invocation_context.context = get_event_request_context(invocation_context)
        try:
            payload = self.request_templates.render(invocation_context)
        except Exception as e:
            LOG.warning("Failed to apply template for SNS integration", e)
            raise
        uri = (
            invocation_context.integration.get("uri")
            or invocation_context.integration.get("integrationUri")
            or ""
        )
        region_name = uri.split(":")[3]
        headers = aws_stack.mock_aws_request_headers(service="sns", region_name=region_name)
        return make_http_request(
            config.service_url("sns"), method="POST", headers=headers, data=payload
        )
class LambdaProxyIntegration(BackendIntegration):
    @classmethod
    def update_content_length(cls, response: Response):