### 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`
invocation_context.response = response
            response_templates = ResponseTemplates()
            response_templates.render(invocation_context)
            invocation_context.response.headers["Content-Length"] = str(len(response.content or ""))
            return invocation_context.response
        raise Exception(
            f'API Gateway integration type "{integration_type}", action "{uri}", method "{method}"'
        )
    elif integration_type == "AWS":
        if "kinesis:action/" in uri:
            if uri.endswith("kinesis:action/PutRecord"):
                target = kinesis_listener.ACTION_PUT_RECORD
            elif uri.endswith("kinesis:action/PutRecords"):
                target = kinesis_listener.ACTION_PUT_RECORDS
            elif uri.endswith("kinesis:action/ListStreams"):
                target = kinesis_listener.ACTION_LIST_STREAMS
            else:
                LOG.info(
                    f"Unexpected API Gateway integration URI '{uri}' for integration type {integration_type}",
                )
                target = ""
            try:
                invocation_context.context = helpers.get_event_request_context(invocation_context)
                invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
                request_templates = RequestTemplates()
                payload = request_templates.render(invocation_context)
            except Exception as e:
                LOG.warning("Unable to convert API Gateway payload to str", e)
                raise