### 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`
response = invocation_context.response
    integration = invocation_context.integration
    int_responses = integration.get("integrationResponses") or {}
    if not int_responses:
        return response
    entries = list(int_responses.keys())
    return_code = str(response.status_code)
    if return_code not in entries:
        if len(entries) > 1:
            LOG.info("Found multiple integration response status codes: %s", entries)
            return response
        return_code = entries[0]
    response_params = int_responses[return_code].get("responseParameters", {})
    for key, value in response_params.items():
        
        if str(key).lower().startswith("method.response.header."):
            header_name = key[len("method.response.header.") :]
            response.headers[header_name] = value.strip("'")
    return response
def invoke_rest_api_from_request(invocation_context: ApiInvocationContext):